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Energy 149 (2018) 990e999

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Energy

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Minimising the effects of manufacturing uncertainties in MEMSEnergy harvesters

H. Madinei*, H. Haddad Khodaparast, M.I. Friswell, S. AdhikariCollege of Engineering, Swansea University, Bay Campus, Fabian Way, Crymlyn Burrows, Swansea, SA1 8EN, United Kingdom

a r t i c l e i n f o

Article history:Received 16 August 2017Received in revised form8 February 2018Accepted 10 February 2018Available online 14 February 2018

Keywords:UncertaintyMEMSEnergy harvesterPiezoelectricNonlinear dynamics

* Corresponding author.E-mail address: hadi.madinei@swansea.ac.uk (H. M

https://doi.org/10.1016/j.energy.2018.02.0480360-5442/© 2018 Elsevier Ltd. All rights reserved.

a b s t r a c t

This paper proposes the use of an electrostatic device to improve the performance of MEMS piezoelectricharvesters in the presence of manufacturing uncertainties. Different types of uncertain parameters havebeen considered and randomised according to their experimentally measured statistical properties. It hasbeen demonstrated that manufacturing uncertainty in MEMS harvesters results in a lower output power.Monte Carlo Simulation is used to propagate uncertainty through the MEMS mathematical model. It hasbeen found that the uncertainty effects can result in two sets of samples. The first set of samples arethose with resonance frequency higher than nominal values and the second set includes samples withresonance frequencies lower than the nominal value. The device proposed in this paper can compensatefor the effects of variability in the harvester by tuning the resonance frequency to the nominal design.This device is composed of a symmetrical arrangement of two electrodes, which decrease the resonancefrequency from its nominal value. However, achieving precise symmetrical conditions in the device on amicro-scale is not feasible. Therefore, the effects of an unsymmetrical arrangement due to manufacturingvariability are also investigated. The device includes two arch-shaped electrodes that can be used toincrease the resonance frequency.

© 2018 Elsevier Ltd. All rights reserved.

1. Introduction

In our environment, there are a variety of ambient energysources such as solar, temperature, wind and vibration. Convertingwaste ambient energy into small amounts of electrical energy canpower many useful low energy consuming Micro Electro Mechan-ical systems (MEMS) in different applications ranging fromwirelesssensor networks tomedical implants [1e5]. Using vibration sourceshas gained more popularity due to their high availability in variousenvironments. There are different transduction methods that canbe used to convert mechanical vibrations into electrical energy. Themost common types of transduction mechanisms are electrostatic,electromagnetic and piezoelectric. Electrostatic energy harvesterscan convert ambient vibrations into electrical energy using acharged variable capacitor. The capacitance changes based on themechanical vibrations. Any change in capacitance results in acharge rearrangement of the electrodes of the capacitor and,consequently, the charge flows through the electrical circuit [6].

adinei).

Electromagnetic vibration-based energy harvesters usually consistof a coil, a permanent magnet and a suspension spring. Accordingto the Faraday law of electromagnetic induction, when the coilexperiences a change in the magnetic flux because of a relativemotion between the magnet and the coil, electrical energy isgenerated [7]. In piezoelectric energy harvesters, the mechanicalenergy of vibrations is transformed into electrical energy bypiezoelectric material. When piezoelectric material is deformed,the central molecules in the crystal become polarized and form adipole. If the dipoles are arranged suitably, then two of the materialsurfaces become positively and negatively charged [8]. Due to thecompatibility between piezoelectric material deposition and theMEMS fabrication process, piezoelectric convertors have beenrecognized as offering more benefits [9].

Generally, in vibration-based energy harvesters (VBEHs),maximum energy can be harvestedwhen the harvester is excited atits resonance frequency. In most cases, the resonance frequency ofthe harvester does not match the frequency of the vibration source,and as a result the output power decreases significantly. There aretwo methods to increase the efficiency of the VBEHs: broadeningthe bandwidth, and tuning the resonance frequency of theharvester [10,11].

Nomenclature

bwp Electromechanical couplingFe Electrostatic forcebj Stochastic parametersMt Tip massz0 Amplitude of base excitationbU Frequency of base excitationFf Follower forcebca Damping of the harvesterε0 Permittivity of free spacem Damping coefficientCp Internal capacitance of the piezoelectric layerR Load resistanceεs33 Permittivity component at constant strain2a Angular of overlap between the fingersgr Radial gap between the comb fingersg01 Air gap between the micro-beam and the straight

electrodeg02 Air gap between the micro-beam and the straight

electrodee31 Equivalent piezoelectric coefficientLc Length of piezoceramic layersvp Voltage across each piezoceramic layer

H. Madinei et al. / Energy 149 (2018) 990e999 991

In the last few years, many studies have focused on thesemethods to improve the harvested power of VBEHs. Masana andDaqaq [12] developed an electromechanical nonlinear model of anaxially loaded energy harvester. They illustrated that the axial staticload can be used to tune the system over a wide range of fre-quencies. Challa et al. [13] investigated a vibration energy har-vesting device with autonomously tunable resonance frequency.They used a piezoelectric cantilever beam array with magnetsattached to the free ends of cantilever beams to tune the systemresonance frequency bymagnetic force. Miller et al. [14] proposed apassive self-tuning beam resonator with sliding proof mass alongthe beam. This model enables the energy harvesting system toadjust the natural frequency of the system and thereby increase theenergy harvested over time. Erturk and Inman [15] investigatedbroadband high-energy orbits in a bistable piezomagnetoelasticenergy harvester over a range of excitation frequencies. Malaji andAli [16] analysed broadband energy harvesting using multiplelinear harvesters. They showed that the bandwidth of harvestingcan be increased by using an array of coupled pendulums withmechanical grounding. In addition, they found that the bandwidthand the total harvested power saturates with the number ofpendulums.

A mismatch between resonance frequency and vibration sourcefrequencymay exist either due to changes inworking conditions, orexcessive manufacturing tolerances and errors. In MEMS devicesdue to fabrication processes such as mask alignment, deposition,photolithography, etching and drying, manufacturing tolerancesare generally high and in some cases, they can be higher than ±10%of nominal values [17]. Therefore, parameter uncertainty cansignificantly affect the performance of MEMS devices. Uncertaintyanalysis of MEMS devices has been studied by several authors ofprevious studies. Agarwal and Aluru [18] presented a framework toquantify different kinds of outputs in MEMS structures such asdeformation and electrostatic pressure in these devices. Agarwal

and Aluru [19] proposed a framework to include the effect of un-certain design parameters of MEMS devices. Based on this frame-work they investigated the effect of variations in Young's modulus,induced because of variations in the manufacturing process pa-rameters or heterogeneous measurements, on the performance of aMEMS switch.

In this paper, an electrostatic device is proposed in order tocompensate for the effect of manufacturing uncertainties on theperformance of MEMS piezoelectric harvesters. In this model, theresonance frequency of the harvester is tuned using an archelectrode and two straight electrodes. Manufacturing uncertaintycould potentially change the harvester's resonance frequency andconsequently the deviation from its nominal value may be posi-tive or negative. Therefore, there is a need to tune the harvester'sresonance frequency to a higher (hardening) or lower (softening)frequency. By applying voltage to the aforementioned electrodes,the resonance frequency of the harvester can be adjusted throughhardening and softening mechanisms. Applying DC voltage to thearch shaped electrode creates a tensile follower force which canincrease the resonance frequency of the harvester linearly.Conversely, the resonance frequency of the harvester can bedecreased by applying voltage to the straight electrodes, whichcreates a softening nonlinearity. The Galerkin method has beenused to discretise the equations of motion. Only a single modeapproximation was used in the simulations. This is mainly basedon the assumption that the excitation frequencies are close to thesystem's first resonance frequency. The problem considered inthis paper is non-linear due to the electrostatic forces. Further-more, uncertainty in the model parameters is considered;therefore, this is a dynamic problem with the effects of bothnonlinearities and uncertainties. Such problems have notreceived significant attention in the literature. In the presence ofuncertainty, a semi-analytical solution such as Harmonic Bal-anace (HB) or Incremental Harmonic Balance (IHB) used by au-thors in previous papers [20,21] cannot be applied. This isbecause it is not feasible to perform a convergence study on therequired number of truncated terms of nonlinear force forthousands of samples generated by Monte Carlo Simulation. Thispaper, for the first time to our knowledge, demonstrates the useof a shooting method in conjunctionwith Monte Carlo Simulation(MCS) to solve a nonlinear uncertain problem. The shootingmethod may be considered to be more efficient than a timeintegration method for uncertainty propagation. This paperalso shows how nonlinear electrostatic forces, can be used toimprove the performance of MEMS devices in the presence ofmanufacturing uncertainties.

2. Model description and mathematical modelling

Fig. 1 shows the proposed model in this paper. The model is anisotropic micro-beam of length L; width a; thickness h; density r

and Young's modulus E; sandwiched with piezoceramic layershaving length Lc, thickness h0, Young's modulus E0 and density r0throughout the micro-beam length and located between twostraight-shaped electrodes and one arc-shaped electrode. As illus-trated in Fig. 1, the piezoceramic layers are connected to theresistance ðRÞ and the coordinate system is attached to the middleof the left end of the micro-beam, where x and z refer to the hor-izontal and vertical coordinates respectively. The free end of themicro-beam is attached to the two arc-shaped comb fingers, whichsubtend angle a at the base of the beam and remain parallel to thefixed arc-shaped electrode. The governing equation of transversemotion can be written as [22].

Fig. 1. Schematic of the proposed energy harvester.

H. Madinei et al. / Energy 149 (2018) 990e999992

v2

vbx2

0@EI�bx; bj� v2bw

vbx2

1Aþ rA�bx; bj� v2bw

vbt2 þ bcavbwvbt þ Ff

�bj� v2bwvbx2

� bw�bj�v�bt� dd�bx�dbx �

dd�bx�Lc

�dbx

!

¼ Fe

�bj�þ z0 bU2�rA�bx; bj�þMtd

�bx � L��

cos�bUbt� (1)

and subjected to the following boundary conditions

bw�0;bt� ¼ 0;v bw�0;bt�

vbx ¼ 0

v

vbx0BB@EI

�bx; bj� v2 bw�L;bt�vbx2

1CCA ¼ Mt

0BB@v2 bw�L;bt�vbt2

1CCAv2 bw�L;bt�

vbx2 ¼ 0

(2)

where bj denotes the stochastic parameters, which have been usedas an input to the mathematical model.

In Equation (1), bw is the transverse deflection of the beamrelative to its base at the position bx and time bt ; bca is the viscous airdamping coefficient, dðbxÞ is the Dirac delta function, zðbtÞ is the baseexcitation function, Ff is the follower force which is applied to theharvester by the arc-shaped electrode, Fe is the electrostatic forcewhich is applied to the harvester by the straight-shaped electrodes,vðbtÞ is the voltage across the electrodes of each piezoceramic layer,bw is the coupling term which is dependent on the type of connec-tion between the piezoceramic layers (i.e. series or parallel con-nections). By considering the parallel connection between these

v2

vx2

s1ðx;jÞ

v2w

vx2

!þ s2ðx;jÞ

v2w

vt2þ ca

vwvt

þ�af ðjÞV2

2

� v2wvx2

� wpðjÞv

¼ aeðjÞV21

HðxÞ

ð1�wÞ2� HðxÞðrðjÞ þwÞ2

!þ ðs1s2ðx;jÞ þ s2dðx� 1ÞÞU2 c

layers and Kirchhoff's laws, the electrical circuit equation can beexpressed by

Cp

�bj� dvp

�bt�dbt þ

vp

�bt�2R

þ ip

�bt ; bj� ¼ 0 (3)

where the internal capacitance ðCpÞ; coupling term ðbwpÞ and thecurrent source can be obtained as [22].

Cp

�bj� ¼ εs33aLch0

;

bwp

�bj� ¼ e31ah0

�h0 þ h

2

�2

� h2

4

!;

ip

�bt ; bj� ¼ e31a2

ðh0 þ hÞZLc0

v3bw�bx;bt�vbx2

vbt dbx(4)

In Equation (4) e31 is the equivalent piezoelectric coefficient andεs33 is the permittivity component at constant strain with the planestress assumption for the beam. Using electrostatic principles, theelectrostatic force between the micro-beam and the straight elec-trodes ðFeÞ can be written as [23].

Fe

�bj� ¼ε0a H

�bx�2

0BBB@ V21�

g01�bw�2 � V21�

g02þbw�2

1CCCA (5)

where

H�bx� ¼ H

�bx � bd1

�� H

�bx � bd2

�(6)

In Equation (5), ε0 is the permittivity of free space, HðbxÞ is theHeaviside function, V1 is the applied DC voltage to the straight elec-trodes, g01 and g02 are the air gaps between themicro-beamand thestraight electrodes. By applying voltage to the arc-shaped electrodeshown in Fig. 1, the amplitude of Ff can be tuned. Fig. 2 shows for asmall angular deflection of the micro-beam ðqÞ; the angular overlapbetween the fingers and the arched-shaped electrode is always 2aand the force remains a follower force in all conditions.

Based on electrostatic principles, the amplitude of the followerforce ðFf Þ can be written as

Ff

�bj� ¼ ε0ag2r

V22 La (7)

where gr is the radial gap between the comb fingers and arc-shapedelectrode, a is thewidth of the resonator, L is the length of the beam(the thickness of the arc-shaped electrode can be ignored), and V2 isthe applied voltage to the arched-shaped electrode. For conve-nience, Equations (1) and (3) can be re-written in a non-dimensional form as follows

pðtÞ�ddðxÞdx

� ddðx� lcÞdx

�

osðUtÞ(8)

Fig. 2. A micro-beam with an arc-shaped comb fingers in its (a) unperturbed state and (b) perturbed state.

H. Madinei et al. / Energy 149 (2018) 990e999 993

dvpðtÞdt

þ lðjÞvpðtÞ þ gðjÞZlc0

v3wvx2vt

dx ¼ 0 (9)

where

w ¼ bwg01; x ¼ bxL; t ¼ btT; U ¼ T bU; lc ¼ Lc

L

ca ¼ bcaL4ðEIÞC T ; T ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðrAÞCL4ðEIÞC

s; wp ¼

bwpL2

ðEIÞCg01; af ¼ ε0aL3a

ðEIÞCg2r

ae ¼ ε0aL4

2g301ðEIÞC; s1 ¼ z0L

4 ðrAÞCT2ðEIÞCg01

; s2 ¼ z0L3Mt

T2ðEIÞCg01; l ¼ T

2RCp;

g ¼ e31ag01ðh0 þ hÞ2CpL

; r ¼ g02g01; s1 ¼

EI�bx;j�ðEIÞC ; s2 ¼

rA�bx;j�

ðrAÞC(10)

where ðEIÞC and ðrAÞC are related to the bending stiffness and massper unit length of the beam with piezoelectric layers. To eliminatethe spatial dependence in Equations (8) and (9) the Galerkindecomposition method is used. The deflection of the micro-beamcan be represented as a series expansion in terms of the eigen-functions of the micro-beam, i.e.

wðx; tÞ ¼XNi¼1

UiðtÞ4iðxÞ (11)

where 4iðxÞ is the ith linear undamped mode shape of the straightmicro-beam and UiðtÞ is the ith generalized coordinate. Equations(8) and (9) can be converted into a system of differential equationsusing this method. A single-mode approximation yields thefollowing equations

€UðtÞ þ 2mun _UðtÞ þ u2nUðtÞ � qp vpðtÞ

¼ at

Z10

V21HðxÞ

ð1� U4ðxÞÞ2 �V21HðxÞ

ðr þ U4ðxÞÞ2!4dxþ FU2 cosðUtÞ

(12)

_vpðtÞ þ lvpðtÞ þ b _UðtÞ ¼ 0 (13)

where

M¼Z10

s2ðx;jÞ42ðxÞdx;C¼ ca

Z10

42ðxÞdx;Kf ¼af V22

Z10

4ðxÞ4IIðxÞdx;

Km¼Z10

�sII1ðx;jÞ4IIðxÞþ2sI1ðx;jÞ4IIIðxÞþ s1ðx;jÞ4IV ðxÞ

�4ðxÞdx;

F¼ 1M

0@s1

Z10

4ðxÞs2ðx;jÞdxþs2

Z10

4ðxÞdðx�1Þdx1A; at ¼ae

M;

b¼ g

�d4ðlcÞdx

�; m¼ C

2Mun; un¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKmþ Kf

M

r; qp¼wp

M

�d4ðlcÞdx

�(14)

Due to the electrostatic nonlinearity in Equation (8), finding ananalytical solution to study the dynamic behavior of the system isquite complicated. However, there are different methods to find anapproximate analytical solution of Equations (8) and (9). Previously,the authors (Madinei et al. [20]) have used the harmonic balancemethod to study the dynamic behavior of the system by consid-ering an approximate electrostatic force using a Taylor expansion.They assumed a symmetric electrostatic force in their approxima-tion and showed that acceptable convergence can be obtained byincluding terms up to ninth-order. However, in the presence ofmanufacturing uncertainties, the electrostatic force could beunsymmetric due to the variabilities in the air gap, and more termsmay need to be included to reach acceptable convergence. There-fore, using the harmonic balance method makes the uncertaintypropagation tedious because for every different sample, the num-ber of truncated terms should be determined. In this study, theshooting method [24] is used to investigate the dynamic behaviorof the system. Generally, the shooting method is a powerful anduseful method to find periodic solutions to a nonlinear system, andit is computationally more time efficient than direct integrationmethods. The shooting method can also find unstable solutionsalthough this is not needed for the analysis undertaken in thispaper. By introducing X1¼U;X2¼ _U andX3¼ vp; Equations (12) and(13) can be rewritten as

Table 2Most sensitive parameters to manufacturing uncertainties [17].

Data Mean Std COV (%)

Thickness, h (mm) 4 0.35 8.75Thickness, h0 (mm) 2 0.175 8.75Young’s modulus, E (GPa) 169.6 16.58 9.78Young’s modulus, E0 (GPa) 65 6.35 9.78Air gap, g0 (mm) 40 2.52 6.3Air gap, gr (mm) 3 0.18 6.3

H. Madinei et al. / Energy 149 (2018) 990e999994

_X1 ¼ X2; (15)

_X2 ¼ F cosðUtÞ � 2munX2 � u2nX1 þ qpX3

þ at V21

Z10

HðxÞ

ð1�X14ðxÞÞ2� HðxÞ

ðrþX14ðxÞÞ2

!4 dx

(16)

_X3 ¼ �lX3 � bX2 (17)

To find a periodic solution to Equations (15)e(17), an appro-priate set of initial conditions ðh1; h2; h3Þ must be identified. Toproceed with the shooting technique, for convenience, thefollowing variables are defined:

X4 ¼ vX1vh1

;X5 ¼ vX1vh2

;X6 ¼ vX1vh3

;

X7 ¼ vX2vh1

;X8 ¼ vX2vh2

;X9 ¼ vX2vh3

;

X10 ¼ vX3vh1

;X11 ¼ vX3vh2

;X12 ¼ vX3vh3

(18)

The shooting technique requires the simultaneous integration ofEquations (15)e(17) plus the time derivatives of the variables ð _X4 �_X12Þ in the time domain for one period of excitation. The initialconditions for solving the set of differential equations are defined as

X1ð0Þ ¼ h10; X2ð0Þ ¼ h20; X3ð0Þ ¼ h30; X4ð0Þ ¼ 1;

X5ð0Þ ¼ 0;X6ð0Þ ¼ 0;X7ð0Þ ¼ 0; X8ð0Þ ¼ 1;

X9ð0Þ ¼ 0;X10ð0Þ ¼ 0;X11ð0Þ ¼ 0;X12ð0Þ ¼ 1

(19)

h10; h20 and h30 are initial guesses for the initial condition thatresult in a periodic solution. Generally, these initial guesses deviatefrom the exact values by an error or correction dh: By calculating thevalues of X4 � X12 at one period and substituting them in thealgebraic system of equations below, the error can be found for eachset of initial guesses [24].2424 X4

X7X10

X5X8X11

X6X9X12

35� ½I�3524 dh1

dh2dh3

35¼24 h10 � X1ðT ; h10; h20;h30Þh20 � X2ðT ; h10; h20;h30Þh30 � X3ðT ; h10; h20;h30Þ

35 (20)

By trying different initial guesses and using Equation (20), theerror ðdh1; dh2 and dh3Þ can be minimized and convergence isachieved. Then, the peak power through the resistance can be ob-tained by substituting vp into the following equation

Table 1Geometrical and material properties of the harvester.

Length, L ðmmÞ 3000Width, a ðmmÞ 1000Thickness, h ðmmÞ 4Thickness, h0 ðmmÞ 2Young's modulus, E (GPa) 169.6Young's modulus, E0 (GPa) 65Air gap, g0 ðmmÞ 40Air gap, gr ðmmÞ 3Damping coefficient, ca(N.s/m) 0.002Density of Si beam, r (kg/m3) 2330

Density of PZT, r0(kg/m3) 7800

Equivalent piezoelectric coefficient, e31 ðCm�2Þ �11.18

Permittivity component, εs33(nF/m) 13.48

P0 ¼ v2pR

(21)

3. Numerical results and discussion

To demonstrate the analysis presented in the previous section, abimorph piezoelectric micro cantilever beam is considered withthe geometrical and material properties as listed in Table 1.

Based on the experimental results [17], the most importantvariations in the fabrication parameters include thickness, air gapand Young's modulus. To show the effect of these parameters on theperformance of the harvester, a Gaussian distribution of parametersis assumed and given in Table 2. There are generally two modelsthat can be used to represent parametric uncertainties, that may becategorized into two groups: (i) probabilistic and (ii) non-probabilistic. There are several models within these two cate-gories for modelling uncertainty in numerical models. Details ofthese models and methods of uncertainty propagation throughnumerical model can be found in Refs. [25,26]. However, these arenot the concern of this paper.

Considering the mean parameters of the micro-beam, theoptimal resistance of the harvester is obtained at its resonancefrequency. As shown in Fig. 3a, by exciting the harvester at itsresonance frequency, 12.6 nW power can be harvested at theoptimal resistance. In addition, as Fig. 3b shows, the maximumdeflection of the beam at the resonance frequency is less than40mm: To investigate the effect of the manufacturing uncertaintieson the performance of the MEMS piezoelectric harvester, differentnumbers of samples are generated, and the Monte Carlo simulationis used for uncertainty propagation. Fig. 4a shows that the Proba-bility Density Function (PDF) of the power does not significantlychange when the number of samples is increased from 1500 to2000, hence 2000 samples will be enough for uncertainty analysis.

Due to the variability of the parameters, there is a large devia-tion in the resonance frequencies of the samples, and this cansignificantly decrease the performance of the harvester. Fig. 4bshows that the mean resonance frequency of the harvester is406 Hz, however, there are many samples which have resonancefrequencies either greater or less than the mean value. On the otherhand, because of this variability in resonance frequency, the har-vested power of most samples deviates from the power of thesystem with the mean parameters when it is excited at the mean/nominal resonance frequency (see Fig. 4a). In order to compensatefor the effect of manufacturing uncertainties, the resonance fre-quency of samples can be adjusted by applying voltage to theelectrodes. Fig. 5a shows that by applying voltage to the straightelectrodes, the resonance frequency of the micro-beam decreasesdue to the softening nonlinearity of the electrostatic field. Consid-ering this nonlinearity, there are multiple solutions for the micro-beam response within the frequency range close to the frequencyof the vibration source. In order to harvest more power, the microbeam response should be at the higher of the two solutions and

Fig. 3. (a) Variation of the piezoelectric peak power with load resistance at nominal resonance frequency (b) Displacement frequency response curve with the optimal resistance.

Fig. 4. Probability density function of (a) harvested power and (b) resonance frequency.

H. Madinei et al. / Energy 149 (2018) 990e999 995

close to the resonance frequency. However, being at the highersolution depends on the initial conditions and therefore theresponse at the higher amplitude cannot be guaranteed. Adjustingthe applied DC voltage can ensure that the response of theharvester will be in the higher solution. For a given excitation fre-quency if the harvester response happens to be in the loweramplitude solution the DC voltage is increased until a region isreached where the harvester only has a single solution. The DCvoltage is then slowly reduced and the harvester follows highamplitude solution until the resonance is obtained.

As shown in Fig. 5a, by applying 8 V to the electrodes, theresonance frequency of the sample is decreased by 7.5% to matchthe frequency of vibration source. Consequently, the harvestedpower can be increased by 13.2 nW. Fig. 5b shows that the reso-nance frequency of the micro-beam can be increased by applying afollower force. In Fig. 5b, an arbitrary sample with a resonancefrequency less than 406Hz has been considered. Using the hard-ening mechanism and applying 13.6 V, the resonance frequency ofthe sample can be increased by 7.7% and therefore more power canbe harvested.

In both mechanisms, the resonance frequency of the sample istuned based on the electrostatic force. The magnitude of this forcecan be controlled by voltage, air gap and overlapping area betweenelectrodes. Generally, the air gap and overlapping area areconsidered to be designed parameters and they are constant.However, based on Table 2, the air gaps between electrodes will beaffected by manufacturing uncertainties. Therefore, depending onthe air gaps between electrodes, the resonance frequency of asample can be tuned by applying DC voltage. In the hardeningmechanism, by applying voltage to the arch-shape electrodes, theresonance frequency of the harvester is changed linearly. However,due to the geometric configuration of the electrodes, in the soft-ening mechanism the behavior of the harvester is affected byelectrostatic nonlinearity.

In addition, due to the variabilities in the air gap between twostraight electrodes the system may become unsymmetrical. Ac-cording to Equation (5), the amplitude of the electrostatic force canbe controlled by applying a DC voltage ðV1Þ and changing the airgap between electrodes. As shown in Fig. 6a, by considering equalinitial gaps between electrodes ðg01 ¼ g02 ¼ 40mmÞ; the resonance

Fig. 5. Tuning resonance frequency of micro beam using (a) softening ðd2 � d1 ¼ 0:5LÞ and (b) hardening ða ¼ 30� Þ mechanism.

Fig. 6. (a) Tuning resonance frequency of symmetrical model ðg01 ¼ g02 ¼ 40mmÞ (b) comparison of symmetrical and unsymmetrical model ðg01 ¼ 44:6mm; g02 ¼ 36mmÞ.

H. Madinei et al. / Energy 149 (2018) 990e999996

frequency of the sample can be tuned to the nominal frequency byapplying 10.1 V to the electrodes. However, by including the vari-abilities in the air gaps, the resonance frequency for the givensample may be changed by applying 8 V to the electrodes (SeeFig. 6b). Therefore, in comparison with the symmetrical model,depending on the initial gaps between electrodes in the unsym-metrical model, the applied DC voltage may either be increased ordecreased. In addition, as shown in Fig. 7, the output power due tothe steady state response for the unsymmetrical model can bedifferent in comparison with the symmetrical model. In the un-symmetrical model, due to the nonzero static deflection, the outputvoltage will be affected by a DC offset. Therefore, the output voltagewill swing between two different values, instead of the usual þvacand �vac: Consequently, there will be double peaks in the steadystate response of the output power in the unsymmetrical model(see Fig. 7a).

Generally, in nonlinear energy harvesters, maximum power canbe harvested when the harvester responds on the upper branch inthe vicinity of its resonance frequency. For any changes in the initialcondition, the harvester tends to jump down to the lower branch,

thereby decreasing harvested power significantly. As shown inFig. 8a, by jumping down from point P2 in Fig. 6b the harvestedpower decreases by 89%.

Fig. 8b shows that in the case of jumping down to the lowersolution ðP0Þ; the applied DC voltage is increased until a region isreached where the harvester only has a single solution. Then bydelivering a gradually decreasing voltage in the fixed frequencydirection, the harvester follows the high amplitude solution untilresonance is obtained.

The voltage source in both symmetrical and unsymmetricalmodels can be charged through the harvested power from theelectrostatic side. Generally, electrostatic harvesters require anenergy cycle to convert mechanical energy to electrical energy [27].The energy conversion cycles mostly rely on charge or voltageconstraint concepts. In both cycles, electrical charge is stored in avariable capacitor when its capacitance is high. Then the capaci-tance of the capacitor is reduced by mechanical vibration, andeventually the capacitor will be discharged.

Considering the voltage constraint cycle, as shown in Fig. 9a,there are two variable capacitors between the beam and the

Fig. 7. Output power due to the steady state response at 406Hz (a) unsymmetrical model ðV1 ¼ 8VÞ (b) symmetricalðV1 ¼ 10:1VÞ.

Fig. 8. (a) The output power of the harvester on the lower branch ðU ¼ 406 Hz;V1 ¼ 8 VÞ (b) Moving from the lower branch to the higher one by decreasing the voltage.

Fig. 9. (a) Variable capacitors in the proposed model (b) Electrical circuit.

H. Madinei et al. / Energy 149 (2018) 990e999 997

Fig. 10. Harvested power of samples based on (a) softening and (b) hardening mechanism.

Fig. 11. (a) Applied DC voltage versus harvested power for softening mechanism and (b) hardening mechanism.

Fig. 12. Harvested power of samples before and after applying DC voltage.

H. Madinei et al. / Energy 149 (2018) 990e999998

straight electrodes. In each cycle of vibration, these capacitors arecharged and discharged continuously and they can charge thevoltage source ðV1Þ based on the voltage constraint cycle. Therefore,in both symmetrical and unsymmetrical models the voltage sourceis self-chargeable and the harvested power from the electrostaticside is used to keep the voltage source constant [20].

Considering all samples, as shown in Fig. 10, by applyingdifferent voltages to the electrodes the resonance frequency of thesamples matches the excitation frequency and more power can beharvested. Fig. 11 shows that by applying DC voltage to the straightelectrodes ðV1Þ up to 26.6 V and the arch-electrodes ðV2Þ up to 24 V,the harvested power of the samples can be improved significantly.Consequently, in comparisonwith Fig. 4a, most samples are shiftedto the region around the power of the system with the mean pa-rameters. As shown in Fig. 11a, the mean applied DC voltage to thestraight electrodes (point A) is 8.5 V and in most cases the har-vested power of the samples is close to the power of the systemwith the mean parameters.

However, for the hardening mechanism, the mean applied DCvoltage (point B in Fig. 11b) is 12.2 V, which is 3.7 V greater than the

H. Madinei et al. / Energy 149 (2018) 990e999 999

mean applied DC voltage for the softening mechanism. Further-more, the mean harvested power of the samples for the hardeningmechanism is less than 12 nW. Therefore, the electrostaticnonlinearity in the softening mechanism can make the tuningmechanism more efficient in comparison with the hardeningmechanism.

In the current analysis, a constant optimal resistance ð70 kUÞ isused for all samples; however this resistance can be optimized foreach sample [20]. In addition, since the axial deflection of the beamis negligible, the power loss of voltage source V2 is small. Consid-ering the results of both mechanisms, as shown in Fig. 12 by usingthe electrostatic force in both mechanisms, the effect ofmanufacturing uncertainties can be compensated and after tuningthe resonance frequencies of the samples, the harvested power ofthe samples varies between 8 and 14 nW.

4. Conclusions

In this paper, the effect of manufacturing uncertainties on theperformance of MEMS piezoelectric harvesters was investigated.The steady state solution was obtained by using the shootingmethod and 2000 samples were considered based on the MonteCarlo simulation. From this study, the following important con-clusions were drawn:

- The results showed that variability in a MEMS harvester willsignificantly reduce its performance. This is because the reso-nance frequencies of the samples in most cases deviate from theexcitation frequency and results in lower harvested power. Itshould be noted that the experimental data in the literaturewere used to randomise the model parameters.

- Two tuning mechanisms were proposed in this paper. They canbe used to compensate for the effect of manufacturing uncer-tainty. For each sample depending on its resonance frequency,appropriate DC voltage was applied. Based on these mecha-nisms, it was observed that the harvested power can beincreased by applying DC voltage to the straight electrodes andarch-shaped electrode up to 26.6 V and 24V, respectively.

The problem in this paper is a nonlinear uncertain dynamicproblem. One important challenge with these problems is thecomputational time required to solve them. It is found that usingshooting method in conjunction with MCS is an efficient tool forsolution of these types of problems. Future work will involvedeveloping more efficient methods to solve this type of nonlinearuncertain problem, investigating the role of global sensitivityanalysis on the selection of uncertain parameters, and robustdesign of such a system to passively minimise the adverse effects ofuncertainty in the harvester.

Acknowledgements

HamedHaddad Khodaparast acknowledge the support providedby the EPSRC EP/P01271X/1.

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