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Analytical coupled modeling of a magneto-basedacoustic metamaterial harvesterTo cite this article: H Nguyen et al 2018 Smart Mater. Struct. 27 055010

 

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Analytical coupled modeling of a magneto-based acoustic metamaterial harvester

H Nguyen1, R Zhu1,2, J K Chen1, S L Tracy3 and G L Huang1

1Department of Mechanical & Aerospace Engineering, University of Missouri, Columbia, MO 65211,United States of America2Department of Mechanics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China3WorkSpace Futures, Steelcase Inc., Grand Rapids, MI 49508, United States of America

E-mail: huangg@missouri.edu

Received 8 December 2017, revised 14 February 2018Accepted for publication 26 March 2018Published 20 April 2018

AbstractMembrane-type acoustic metamaterials (MAMs) have demonstrated unusual capacity in controllinglow-frequency sound transmission, reflection, and absorption. In this paper, an analytical vibro-acoustic-electromagnetic coupling model is developed to study MAM harvester sound absorption,energy conversion, and energy harvesting behavior under a normal sound incidence. The MAMharvester is composed of a prestressed membrane with an attached rigid mass, a magnet coil, and apermanent magnet coin. To accurately capture finite-dimension rigid mass effects on the membranedeformation under the variable magnet force, a theoretical model based on the deviating acousticsurface Green’s function approach is developed by considering the acoustic near field and distributedeffective shear force along the interfacial boundary between the mass and the membrane. Theaccuracy and capability of the theoretical model is verified through comparison with the finiteelement method. In particular, sound absorption, acoustic-electric energy conversion, and harvestingcoefficient are quantitatively investigated by varying the weight and size of the attached mass,prestress and thickness of the membrane. It is found that the highest achievable conversion andharvesting coefficients can reach up to 48%, and 36%, respectively. The developed model can serveas an efficient tool for designing MAM harvesters.

Keywords: acoustic metamaterial harvester, vibroacoustic coupling, membrane model, soundabsorption, magnet damping

(Some figures may appear in colour only in the online journal)

1. Introduction

Low-frequency noise (audible sound waves) greatly threatenshuman health and is a major form of wasted energy. Comparedto high-frequency noise, low-frequency noise spreads withmodest attenuation through air, is often able to penetrate thickbarriers with ease, and is not always easy to control. Based onthe conventional mass density law, heavy and massive dis-sipative materials are needed to decrease low frequency soundtransmission; however, lighter weight and compact size arealways more desirable characteristics for real-world applications.New tools to control the propagation of these sound waves in theform of new materials are extremely desirable.

Recently, a membrane-type acoustic metamaterial (MAM)comprised of a thin subwavelength-scale microstructure was

suggested for low-frequency sound attenuation [1–5]. This typeof acoustic metamaterial has attracted large interest in theresearch community because it has a relatively simple geometrycombined with an intriguing capability to dissipate low-fre-quency sound waves based on the local resonant mechanism.Based on the MAM design, the lightweight simple structures[6, 7] which expose broadband attenuation, and pure flexibledesign [8] have been proposed. Compared with conventionalsound attenuation materials typically utilizing thermally-coupleddissipation mechanisms and suffering from inadequate low fre-quency sound attenuation, MAMs can be designed to possessnearly total reflection and/or absorption for targeting low-frequency acoustic sources.

The basic structure of this MAM consists of a prestressedmembrane with one or multiple attached, small, heterogeneous

Smart Materials and Structures

Smart Mater. Struct. 27 (2018) 055010 (10pp) https://doi.org/10.1088/1361-665X/aab993

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masses acting as resonators. The unusual low-frequencyvibroacoustic behavior has been completely characterized, andthe response spectrum shows separate transmission andabsorption peaks around resonances [9, 10]. The low-frequencysound transmission mechanism has also been numericallyexplained through effective mass density and averaged normaldisplacement by using the finite element method, although thenearly total reflection and absorption of the MAM is of limitedfrequency bandwidth.

As one of many potential engineering applications, light-weight MAM honeycomb panels have been designed numeri-cally and experimentally to mitigate low frequency noisespecifically in aerospace structures [11]. For passive acousticmetamaterials, the operative wave frequencies are hardlyadjustable once fabricated. Thus, they cannot adapt to real-lifescenarios under ambient environments. One promising way tomitigate these problems is to incorporate an active element.Chen et al used a gradient magnetic field to actively tune theMAMs [12]. Xiao et al also investigated acoustic properties ofmembrane-type metamaterials actively adopted by externalvoltage to illustrate phase modulation and acoustic waveswitch [13].

In another vein, scientists and engineers have beenstrongly interested in potential applications of acoustic meta-materials in the field of energy harvesting. The unique ability ofacoustic metamaterials to trap acoustic waves in a localizedregime can be used to efficiently harvest sound energy throughintegrating smart elements and electric circuits. From a prac-tical point of view, there is combined value in both protectingpeople from damaging sound as well as actively retrievingenergy from these noises to provide power to electronic deviceswithout the need of batteries. As one of several candidatematerials, piezoelectric materials have been successfully usedin acoustic metamaterials for harvesting structure-borne as wellas air-borne sound energies [14–19]. Although piezoelectricmaterials can generate a relatively high-voltage output fromexternal motion-induced mechanical strain, the high impedancein piezoelectric-based harvesters mandates the load impedanceto be high, which will significantly limit the acoustic energyharvesting efficiency [20–22]. Beside employing piezoelectricmaterials on designing harvesting structures, the voltage-tunable [23] and dielectric elastomer based [24] designs arealso taken a lot of attention. Although these designs show verygood ability on acoustic absorption, the harvested energy andability on harvesting acoustic energy have not mentioned yet.

An alternative way to design an energy harvester is basedon electromagnetic energy conversion, which is suitable for alow impedance and therefore can generate a high output cur-rent [25]. Mikoshiba et al [26] designed an acoustic metama-terial consisting of a spring-loaded magnet enclosed in acapped poly(methyl methacrylate) tube equipped with coppercoils to harvest vibrational energy using electromagneticinduction. Ma et al [4] demonstrated a membrane acousticmetasurface that can convert the acoustic energy to electriccurrent through electromagnetic induction with a power con-version efficiency of 23%, which demonstrates great potentialfor acoustic energy harvesting. However, many design issueshave yet to be resolved and better understood such as

enhancing efficiency, optimizing microstructures, and operat-ing at a frequency bandwidth. A solid analytical model that canaccurately capture the complicated vibro-acoustic-electro-magnetic coupling behavior is greatly needed. The analyticalmethod can provide both computational efficiency and flex-ibility, and therefore can be very useful in the design of multi-functional MAMs for desired engineering applications.

In this paper, we developed a comprehensive vibro-acoustic-electromagnetic coupling model to accurately cap-ture the dynamic behavior as well as the energy absorption,conversion and harvesting in the proposed MAM harvester.Based on the model, sound absorption, acoustic-electronicenergy conversion, and the MAM harvesting coefficient arequantitatively evaluated for various geometries and con-stitutive material properties of the metamaterial. The effi-ciencies of energy conversion and energy harvesting, themaxima of which are 48% and 36%, respectively, are ana-lyzed and optimized by varying the subwavelength-scalemicrostructure and connected circuit configurations. Numer-ical simulations are conducted to validate the analyticalsolutions, and excellent agreement is observed.

2. Analytical modeling of the magneto-basedacoustic metamaterial harvester

Without loss of generality, the magneto-based acoustic meta-material harvester is considered as a MAM with a circular massattached at the center of the membrane, a torus-shape magnetwire attached to the mass, and a permanent magnet placedclose to the magnet wire, as shown in figure 1(a). When theincident pressure wave excites the membrane, the mass andattached wire vibrate in the magnetic field generated by thepermanent magnet, which will induce a current in the movingwire. Thus, an external magnetic force acting on the mass andwire is initiated to prevent the vibration of the mass and magnetwire accordingly. By connecting a resistor, Rl, which mayfunction as electric devices, into the circuit, the acoustic energycan be dissipated through Rl. This energy is called harvestedenergy, and the resistor Rl can be called the external load. Inaddition, the magnet coil also has an internal resistance andhence called internal load denoted by Ri. The electric energydissipated through Ri always becomes heat, and therefore, canbe called wasted energy. By defining the conversion energy asthe total electric energy absorbed by both Ri and Rl, and har-vested energy as the electric energy absorbed by only Rl, theharvested energy is a part of the conversion energy and will bezero when external load is not used (Rl =0).

In our study, we focus on characterizing the MAM soundabsorption in the tube subjected to a plane normal sound waveand investigate the energy harvesting ability of the magneto-based acoustic metamaterial harvester. Perfectly absorbingboundary conditions (BCs) are assumed in both ends of thetube so that there are no multiple reflected waves to theMAM. First, the external magnetic force is obtained analyti-cally by considering the interaction between the magneticfield and the moving electric circuit. Then, the vibro-acoustic-magnetic coupling behavior of the MAM is analyzed through

2

Smart Mater. Struct. 27 (2018) 055010 H Nguyen et al

the modal superposition theory, from which the soundtransmission and reflection of the MAM can be analyticallydetermined. Finally, the energy harvesting ability and itsefficiency are quantitatively estimated by connecting electriccircuits to the wire. This model can also be easily extended toanalyze the MAM-based harvesters with multiple attachedmasses in arbitrary shapes.

2.1. The determination of magnetic force

For a permanent magnet coin, as shown in figure 1(c), themagnetic field around the magnet can be described as [27]

B rM r r

r rV

M n r r

r rS

4d

d , 1

V

V

V

S

S

S

03

3

ò

ò

mp

= ´ ´ -

-

+´ ´ -

-

⎧⎨⎩⎫⎬⎭

( ) ( ) ( )∣ ∣

( ) ( )∣ ∣

( )

where, the magnetization M

is constant in the zdirection.

Since the harvester is axisymmetric, only the magnetic field inthe radial direction affects the motion of the harvester.

Therefore, we rewrite Br as

B r z M

z z

r r rr z

R r z

M

z z

r r rr z z

R z M

z z

r r rr z z

R r z

,4

cos

2 cos z

1 cos d d

4

cos

2 cos

d d4

cos

2 cos

1 cos d d ,

2

r

H

H r s s

s s s S

f s s

H r

H r

s s

s s s S

s s

H r

Hs s

s s s S

f s s

0

0

2

2

22 2 2 3 2

0

0

2

2

2

2 2 2 3 2

0

0

2

2

22 2 2 3 2

f

f

f

f

ò ò

ò ò

ò ò

mp

jj

q j

mp

jj

jmp

jj

q j

=

´-

+ - + -

´ - -

+

-

+ - + -

´ +

´-

+ - + -

´ - -

p

p

p

-

- -

- -

-

-

( )

( )( )

( )

( )

( )∣ ( ) ∣

[ ( )]

( )∣ ( ) ∣

( )∣ ( ) ∣

[ ( )]( )

where, μ0=4π×10−7 N A−2 is the permeability of the free

space; andr

sin ,z

Hf

r

2s

fq =

-⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

where ‘+’ for the first

integral and ‘−’ for the third integral; R, H, rf, and M are theradius, height, fillet radius, and magnetization of the perma-nent magnet, respectively.

When the mass and attached wire move back and forthinside the magnetic field B

with velocity v, the electromotive

force induced inside the circuit reads

tv B l N v B s l

d

dd d , 3

L loop f= - = ´ ⋅ » ´ ⋅

∮ ∮( ) ( ) ( )

where sand N are the tangential vector and the number of the

loops of the wire, respectively.The current density induced inside the connected circuit

is then

jA R R

, 4i l

=

+( )( )

where the external resistor is denoted as Rl and the internal

resistance of the magnet coil is RiN d

d

4 o

c c

2

2=s

with the number of

the loops of the wire being N d

d

2c

w» ( ) and do, dw, dc and σc

are the diameters of the coil, magnet wire, the coil’s cross-section and the electrical conductivity of the magnet wire,respectively.

Due to axisymmetric harvesting, the external distributedforce acting on the wire can then be determined as

f j Bd NB

A R Rn 5b

o r m

i lz

2Wp= ´ = -

+

( )

( )

herein, the radial component of the magnetic field is ignoredsince it will be canceled out in summation; dc, A, v

and mW

are the diameter of the wire, cross-sectional area of the wire,the velocity vector of the mass and its z-directional comp-onent ( vm zW = ), respectively. Thus, the total inducedmagnetic force can be determined by summing the distributed

Figure 1. (a)—The model of the MAM harvester, (b)—themechanical equivalent to the electromagnetic portion of the MAMharvester, (c)—the model of the permanent magnet coin.

3

Smart Mater. Struct. 27 (2018) 055010 H Nguyen et al

forces along the wire as

F f ld NB

R Rnd . 6m b

o r

i lm z

coil

2Wò

p= = -

+

( ) ( )

Since the magnetic force is proportional to the velocity ofthe mass, physically the magnetic force acts exactly like adamping force to the membrane with the magnet damping

coefficient [28] being ,md NB

R Ro r

i l

2

b = p+

( ) as shown in the

figure 1(b). In the frequency domain, the average powerabsorbed by the magnetic damper, therefore, can be calcu-lated as

W F t

d NB

R R

2d

1

2

1

2. 7

d e m e m

o r

i lm m m

0

2

22 2

W

W W

òwpp

b

= ⋅

=+

=

p w

{ } { }

( ) ∣ ∣ ∣ ∣ ( )

Note that, for the system without other energy dissipa-tion, the electromagnetic part of the harvester should beequivalent to a mechanical damper that is proportional to thedamping coefficient βm.

By further analyzing the damping coefficient in the form

of ,md B

d d R N4o r

o c c l

2

2 2b = ps +( )

( )we can conclude that the conver-

sion coefficient depends on the ring diameter and ring cross-sectional diameter, and reaches its maximum value when zeroload is applied.

2.2. Coupling vibro-acoustic-electromagnetic model ofthe MAM

Since the attached mass and wire in the MAM are assumed tobe perfectly bonded to the membrane and rigid compared withthe deformable membrane, to properly capture effects of thefinite mass on the small deformation of the membrane, theproposed MAM is decomposed into two parts: an annulusmembrane and a mass with the attached magnet coil. Con-sidering the elastic membrane here as an elastic plate, thegoverning equation of the annulus pre-tension membrane canbe written as [10, 29]

D T h 0 84 2 2W W Wr w - - = ( )

with the BCs

rr a0, at 9W

W=

¶¶

= = ( )

rV b m b r b0, 2 ¨ at , 10

WWp

¶¶

= = =( ) ( ) ( )

where D h E

12 0 1

3

2s= +n-( ) is the effective bending stiffness

of the membrane, T=σh is the pretension in the membrane,and V b( ) is the shear force at radius b [29]

V b V b Tr

, 11r b

W= +

¶¶ =

( ) ( ) ( )

where V(b), h and ρ are the Kirchhoff’s shear force at theinner boundary, the thickness, and mass density of theannulus membrane, respectively.

The general solution of the equation (8) can be expressedas [29, 30]

r AJ r BY r CI r DK r ,12

0 1 0 1 0 2 0 2W g g g g= + + +( ) ( ) ( ) ( ) ( )( )

where

T

D

h

D2. 131,2

4 4 2 2 42

g a b a a br w

= + = = ( )

J0, Y0 and I0, K0 are the zero-order Bessel functions andmodified Bessel functions, respectively.

Since only the steady-state response field will be con-sidered, the time factor eiωt, which applies to all the fieldvariables, will be suppressed.

By using the BCs, the natural frequencies and modeshape functions of the MAM can be determined by solvingthe eigen-value problem of the matrix

Next, consider a plane sound wave is normally incidenton the MAM. Per the fact that the thickness of the MAM isextremely small compared with the wavelength of low-frequency sound in air, thickness effects of the MAM can beignored. The objective is to determine the acoustic energyreflection, transmission, and absorption within the MAM. Thegoverning equation of the acoustic excited membrane basedon the plate theory and motion equations of the mass can beexpressed as

D T p V r b1 i15

4 2 2W W Wwh rw d+ - - = D + -( ) ( )( )

bV F p S m b2 d , 16mS

2

i

Wòp w- + + D = - ( ) ( )

where Si denotes the circular surface of the mass, and Δp isthe total pressure acting on the membrane

J a Y a I a K aJ a Y a I a K aJ b Y b I b K b

J b J b Y b Y b I b I b K b K b

0. 14

0 1 0 1 0 2 0 2

1 1 1 1 1 1 2 1 2 2 1 2

1 1 1 1 1 1 2 1 2 2 1 2

13

1 1 0 1 13

1 1 0 1 13

1 2 0 2 23

1 2 0 2

g g g gg g g g g g g gg g g g g g g g

g g a g g g a g g g a g g g a g

--

- - - - -

=

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

4

Smart Mater. Struct. 27 (2018) 055010 H Nguyen et al

p p p p P k Z

G r r r r

2 i

d ,

17

i r t z i a

S

0 0W

Wò

w w

d

D = + - = - á ñ +

´ ¢ ¢ ¢

=⎡⎣⎢

⎤⎦⎥

( )∣

( ∣ ) ( )( )

where S is the surface of the MAM; Z0=ρ0c0 is the specificcharacteristic impedance of air; Pi is the amplitude of theincident pressure wave; Wá ñ is the average displacement ofthe membrane; and G r rd ¢( ∣ ) is the deviation of the acousticGreen’s function of the tube measured at z=0 [31]

G r ra

J k r J k r

J k a k k

1, 18

l

rl rl

rl rl a2

1

0 0

02 2 2

ådp

¢ =¢

-=

¥

( ∣ ) ( ) ( )

( )( )

where ka is the wavenumber of air and krl is the lth orderwavenumber of air in the radial direction of the waveguide.

The dissipative property of the membrane is employedhere through the imaginary part of the Young’s modulus,viscosity χ, and effective loss factor η of the membrane as

E EE

i1

, 19m 20

wc hc

n s= + =

+ -( )( )

where E and σ0 are the Young’s modulus and prestress of themembrane.

The superposition method is adopted by assuming thesolution in the form

r r , 20k

k kW Wål=( ) ( ) ( )

whereWk(r) is the kth eigen function of the annulus membrane.Multiplying Wn in equation (15) and conducting an

integral over the surface of the annulus membrane So lead to

T S

h S m b b b

a Zk Z

a

J k r r r J k r r r

J k a k k

a P

i d i

d i

2i2i

d d

2 .

21

k kS

n k o n k k k

Sn o n m n k

n ka

l

Srl n

Srl k

rl rl a

i n

2,

2 2 2

2 2

20

02

1

0 0

02 2 2

2

o

o

W W

W W W W

W W

W W

W

ò

ò

ò òå

l wh d w w hw w

r wb

p wwp

p

å + + -

´ + +

+ á ñá ñ -

´¢ ¢ ¢

-

= á ñ

=

¥

⎧⎨⎪

⎩⎪⎡⎣⎢

⎤⎦⎥

⎫⎬⎪

⎭⎪

( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( )

( )

The equation (21) is recognizable as a system of linearequations where unknown variables λk can be numericallydetermined.

Therefore, the far field transmission, reflection coeffi-cients and absorption for radiated plane waves of the MAMcan be expressed as

TZ

PR T A R T

i1 1 ,

22i k

k k0 2 2Wåw

l= á ñ = - = - -∣ ∣ ∣ ∣

( )respectively.

For the acoustic energy characterization, the acousticinput energy is calculated as

Wa P

Z223i

in

2 2

0

p=

∣ ∣ ( )

and the energy absorbed by the elastic membrane, or elasticabsorption (due to deformable elastic dissipation), which canbe determined by subtracting the energy absorbed by themagnetic damper and the elastic absorption coefficient, arepresented as

W A W W AW

WA

W

W, 24m d m

m din

in in= ⋅ - = = - ( )

respectively. On the other side, the harvested energy throughconnecting the external resistor Rl is

WR

R RW . 25h

l

i ld=

+( )

The acoustic energy conversion coefficient which corre-sponds to the total electric energy (Wd), and harvesting coef-ficient which only corresponds to the amount of electric energyharvested through the external load Rl (Wh) can be defined as

cW

W

W

W

R

R Rc. 26d h l

i lin inj= = =

+( )

3. Results and discussion

To verify the theoretical model of the MAM harvester, severalconfigurations of theoretically-derived acoustic absorptionanalyses are compared with simulation results derived fromthe commercial code, COMSOL Multiphysics. Since themagnetic field generated from the permanent magnet is sta-tionary, a two-step study is employed in the numericalsimulation. In the first step, a ‘Stationary’ study on the‘Magnetic Fields, No Currents’ module is selected to obtainthe static permanent magnetic field. In the second step, a‘Frequency Domain’ on the ‘Acoustic-Structure Interaction’module is performed to study the interaction between acousticdomains and MAM. The magnetic effect is considered byapplying a velocity-dependent body force on the magnet coilcalculated from the magnetic field and the mass’ velocity. TheMAM parameters used in analytical modeling and numericalsimulation are listed in the table 1. The diameter of themagnet wire is set to 0.104 mm, and the internal resistance ofthe magnet coil Ri, therefore, is equal to 0.55Ω.

To validate the theoretical model, the two major resultspredicted by the theoretical analysis and numerical simulationare compared. First, we compare the magnetic fields mea-sured at the center of the magnet coil’s cross-section Br byemploying equation (2) versus COMSOL methods. Theresults for Br are −0.29278 T and −0.29267 T respectively,which shows the tremendous accuracy of the theoreticalmagnet calculation. Second, we compare the wave absorptionof the MAM harvester predicted from the theoretical analysisversus the numerical simulation based on the commercial

5

Smart Mater. Struct. 27 (2018) 055010 H Nguyen et al

code COMSOL Multiphysics as shown in the figure 2. Threecases of the harvester are: (1) viscous membrane (elasticmembrane with dissipative property) but without magneteffect; (2) non-viscous membrane with magnet effect; and (3)both viscous membrane and the magnet effect. The resultsfrom the theoretical analysis and the simulation are nearlyperfectly matched. The developed theoretical analysis there-fore is highly precise and correlates extremely well withnumerical simulation. Furthermore, while the dissipatedenergy from the viscous membrane is very small (less than5%), the absorption of the harvester under the magnet effect issignificantly improved (34%). The proposed harvester con-figuration thus demonstrates a very high potential to harvestenergy. Next, the theoretical analysis is employed to analyzethe parameter effect and further optimize the harvester.

Figures 3(a) and (b) show the effect of the external loadRl on the conversion and harvesting coefficients of the MAMharvester. In the figures, the material parameters of the MAMare the same as those listed in table 1. Through analysis wehave shown that the conversion coefficient, which corre-sponds to the amount of energy absorbed by the circuit,decreases when the external load Rl increases. The best con-version coefficient is observed when the external load is zero.This phenomenon is explained by considering the effect ofthe magnet damping coefficient βm: an increased external loadRl will lower the magnet damping coefficient, which reducesthe dissipated energy through the circuit.

This figure also shows that all the conversion coefficientpeaks are located in the hybridized resonant frequency [4],198 Hz. Due to small elastic dissipative energy (elasticabsorption), the conversion coefficient and the absorptionpeaks of the harvester are at very close frequencies. Theconversion peak here matches with the first absorption peak atthe hybridized resonant frequency, which is very close to thefirst anti-resonant frequency. The first anti-resonance, on theother hand, occurs when the real part of the acoustic Green’sfunction of the MAM [4] is equal to zero and completelydepends on the first two resonant frequencies. The dissipativeproperty of the membrane and the magnet damping coeffi-cient are only related to the resistance of the harvester andwill not contribute to the resonant frequencies, which is alsoconfirmed from the absence of these factors in the char-acteristic equation (14). Therefore, the peak conversioncoefficient in the hybridized resonant frequency will notchange when the external loads are varied. The harvestingcoefficient j of the harvester presented in figure 3(b), whichis defined as the amount of the energy harvested through theexternal load Rl, on the other hand, does not follow the trendof the conversion coefficient. It will initially increase, thenreach the best performance when Rl/Ri ≈1.25, and afterwarddecrease with the increase the external load. The harvestedenergy depends not only on the ratio of external load tointernal load (Rl/Ri) as shown in the equation (25), but alsoon the amount of acoustic energy converted to electric energy.The increase of Rl will lower the magnet damping coefficientβm, and therefore make the conversion energy decrease.Therefore, the increase of the external load Rl does not ensurethe harvested energy is increased as observed in figure 3(b).In addition, since the harvested energy is proportional to theconversion energy (see the equation (26)), the peak of theharvesting coefficient is also located at the hybridized reso-nant frequency.

The next geometric parameter studied is the thickness ofthe membrane h. The thing learnt from the previous discus-sion is the lower external load Rl, which leads to highermagnet damping coefficient βm, the higher conversion coef-ficient c. Therefore, to get the highest conversion coefficient,the external load Rl is set to be zero. The other parameters arekept the same as listed in table 1. When the membranebecomes thicker (or h increases), the effective bending stiff-ness of the membrane D increases, which theoretically makesthe resonant frequencies of the harvester also increase. Con-sequently, the peaks of the conversion coefficient and theelastic dissipative energy through the membrane shift tohigher frequencies as observed in figures 4(a) and (b). Whenthe membrane becomes thicker, the amplitude of the con-version coefficient peak gradually decreases. The amplitudeof the elastic dissipative coefficient peak, in the other hand,gradually increases with the increase of the thickness ofmembrane since the conversion energy partially converts toelastic energy due to the increase of the strain energy [10]. Aswe can expect physically, the effective bending stiffness ofthe membrane will be significantly enhanced with the increaseof the thickness of the membrane, as a consequence, thestored strain energy and the elastic dissipative energy in the

Table 1. The MAM harvester’s parameters.

Element Property Symbol Value

Elasticmembrane

Radius a 25 mm

Thickness h 0.1 mmYoung’s modulus E 1.9×106 PaPoisson’s ratio ν 0.48Mass density ρ 980 kg m−3

Prestress σ0 2.2×105 PaViscosity χ 796 Pa s

Attached mass Radius b 5 mmThickness t 0.35 mmMass density ρm 2900 kg m−3

Permanentmagnet coin

Radius R 3 mm

Height H 3 mmFillet radius rf 0.375 mmMagnetization M 750 kA m−1

Magnet coil Diameter of the coil do 6 mmDiameter of themagnet coil’scross-section

dc 0.4 mm

The gap in betweenthe mass and thepermanent magnet

Δ 0.5 mm

Electricalconductivity

σc 6×107 S m−1

Mass density ρw 8960 kg m−3

6

Smart Mater. Struct. 27 (2018) 055010 H Nguyen et al

Figure 2. Comparisons between the simulation and analytical results for the cases: the membrane with dissipative property but without effectof the magnet, the membrane without dissipative property but with the magnet effect, the membrane with both dissipative property and themagnet effect.

Figure 3. The effect of the external load on the conversion (a) andharvesting (b) coefficients.

Figure 4. The effect of the membrane’s thickness on the conversioncoefficient of the harvester (a) and the elastic dissipative energy ofthe membrane (b).

7

Smart Mater. Struct. 27 (2018) 055010 H Nguyen et al

membrane will be enlarged, as illustrated in figure 4(b). Onthe other hand, a harder membrane sometimes prevents themotion of the mass, which can result in a lower magneticeffect and lowers conversion coefficient as presented infigure 4(a).

The contribution of the mass’ dimensions on the con-version coefficient is presented in the figures 5(a) and (b). Inthe study, the external load Rl is set to be zero and the otherparameters are the same as listed in table 1. It is observed thatwith the increase of the mass radius b or thickness t, theresonant frequencies of the harvester will decrease due toincrease of the mass weight leading to a corresponding shift inthe conversion coefficient peak’s position. While the con-version coefficient peak is clearly lowered when the mass’thickness t is increased, it will initially increase when themass’ radius b increases, then reach the best value atb=2.5 mm, and continually decrease afterwards. Note that,to the best of our knowledge, the best conversion coefficientpeak of 48% observed at b=2.5 mm is much larger thanconversion coefficients possible in a thin MAM. Furthermore,the corresponding absorption peak is nearly 50%, equal to the

limit of the absorption capacity for a thin MAM har-vester [10].

In addition to geometric parameters, we investigated theinfluence of the membrane’s prestress on the conversioncoefficient under free external load as shown in figure 6. Inthis study, all parameters other than the prestress σ0 are thesame as listed in table 1. While the prestress is varied from0.1×105 to 2.0×105 Nm−2, the membrane’s effectivebending stiffness D increases, and the resonant frequencies ofthe harvester thus also increase. The conversion coefficientpeak also shifts to the higher frequency correspondingly. Incontrast to the effects of the membrane’s thickness h on theeffective bending stiffness D (proportional to h3), the pres-tress, which is only linear function of the bending stiffness,can only have a moderate effect on the elastic membrane’sstrain energy and hence the elastic dissipative energy [10].Therefore, the amplitude of the conversion coefficient peak,instead of decreasing continuously like observed fromfigure 4(a) while the prestress increases, gradually reaches abest value of 48% when σ0=0.5×105 N m−2, and thendeceases gradually. It can be understood that the membrane

Figure 5. The effect of the mass’ dimensions on the conversioncoefficient of the harvester.

Figure 6. The effect of the membrane’s prestress on the conversioncoefficient.

Figure 7. The optimized harvesting coefficient of the harvester forσ0=0.5×105 N m−2.

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with an optimized value of prestress leads to a very highmagnet effect, and hence results in the best conversioncoefficient. This state of membrane, therefore, is generallycalled ‘acoustic impedance matching’ [18].

Based on these parameter analyses, we optimized theenergy harvester design. Due to the complex of the multiplevalue optimization, in this study, only two parameters: theexternal load Rl and the radius of the mass b are selected. Inthis study the prestress σ0=0.5×105 Nm−2 is selected.While other parameters are kept the same as listed in thetable 1, two parameters (the mass’ radius and the external loadRl) are varied to find the optimized harvesting coefficient.Figure 7 shows the optimized harvesting coefficientscorresponding to several values of the given mass radius band optimized value of the external load Rl. We observed thatthe maximum harvesting coefficient peak is 36% while thepossible capacity of the harvester is no more than 50%. It canbe realized that with a given value of b, there will have acorresponding optimized value of Rl or Rl/Ri. The resultindicates that increasing b will lower the optimized value ofRl/Ri. Lowering the ratio Rl/Ri makes the portion of theharvested energy in the total electric energy or conversionenergy decreased. As a result, the harvesting coefficient canbe decreased as shown in the figure 7.

In this paper, the MAM harvester is based on axisym-metric model, which can produce elegant analytical solutionsfor both homogeneous and inhomogeneous problems. Basedon the model, the parameter effect and optimization can easilybe conducted. The similar approach can easily be extendedfor other types of structures such as rectangular MAMs orMAMs with multiple attached masses. However, for the caseswith the general geometries, the point-matching methodshould be employed to numerically solve the governingequations and determine the magnet force [9, 10].

4. Conclusion

In this work, an analytical model of a magneto-based MAMharvester is proposed, which shows substantial improvementin all absorption, conversion, and harvesting coefficients. Byinvestigating the contribution of the geometric parameters andmembrane’s prestress on the acoustic performance of theharvester, it is shown that the absorption can reach theabsorption limit of a thin MAM (50%), the conversioncoefficient can reach to 48%; and the harvested coefficientcan also reach to a very impressive number, 36%. Further-more, the theoretical analysis has been well validated bycomparing to numerical simulation results generated using thecommercial code COMSOL Multiphysics. These results lay astrong foundation for developing acoustic metamaterial har-vesters and show tremendous application potential for theproposed MAM harvester.

Acknowledgments

This work was partially supported by Steelcase Inc.

ORCID iDs

G L Huang https://orcid.org/0000-0003-0959-8427

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