Research Article Piezoelectric Wind Energy Harvesting from...

Research ArticlePiezoelectric Wind Energy Harvesting fromSelf-Excited Vibration of Square Cylinder

Junlei Wang123 Sheng Wen4 Xingqiang Zhao2 Min Zhang56 and Jingyu Ran37

1College of Chemical Engineering and Energy Zhengzhou University Zhengzhou Henan 450002 China2Jiangsu Engineering Research Center on Meteorological Energy Using and Control Nanjing University ofInformation Science amp Technology Nanjing 210000 China3Key Laboratory of Low-Grade Energy Utilization Technologies and Systems Chongqing UniversityMinistry of Education of China Chongqing 400044 China4College of Engineering South China Agricultural University Guangzhou Guangdong 510642 China5School of Mechanical Engineering Shanghai Jiao Tong University Shanghai 200240 China6Shipping and Marine Engineering College Chongqing Jiao Tong University Chongqing 40074 China7Power Engineering School Chongqing University Chongqing 400044 China

Correspondence should be addressed to Sheng Wen jlwangzzueducn

Received 4 January 2016 Revised 27 April 2016 Accepted 8 May 2016

Academic Editor Chengkuo Lee

Copyright copy 2016 Junlei Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Self-excited vibration of a square cylinder has been considered as an effective way in harvesting piezoelectric wind energy In presentwork both of the vortex-induced vibration and unstable galloping phenomenon process are investigated in a reduced velocity(119880119903= 119880120596

119899sdot 119863) range of 4 le 119880

119903le 20 with load resistance ranging in 100Ω le 119877 le 1MΩ The vortex-induced vibration covers

presynchronization synchronization and postsynchronization branches An aeroelectromechanical model is given to describe thecoupling of the dynamic equation of the fluid-structure interaction and the equation of Gauss lawThe effects of load resistance areinvestigated in both the open-circuit and close-circuit system by a linear analysis which covers the parameters of the transversedisplacement aerodynamic force output voltage and harvested power utilized to measure the efficiency of the systemThe highestlevel of the transverse displacement and the maximum value of harvested power of synchronization branch during the vortex-induced vibration and galloping are obtainedThe results show that the large-amplitude galloping at high wind speeds can generateenergy Additionally energy can be harvested by utilization of the lock-in phenomenon of vortex-induced vibration under lowwind speed

1 Introduction

Wireless sensor network (WSN) has been widely used invarious fields such as environmental monitoring medicineandmilitaryThebottleneck ofWSN lies in the limited batterylifetime The frequent battery replacement and charging aretime-consuming expensive and environment unfriendly Aninterest in powering the sensors in a self-sustaining way hasbeen increased In literatures the detailed information ofvariousmicropower generators harvesting energy fromwindhuman motions light and temperature difference or theother ways can be found [1ndash6]

Self-excited vibration (SEV) is a common phenomenonin nature and engineering Blevins [7] investigated the mech-anism of the self-excited vibration as shown in Figure 1

In SEV when vortex shedding frequency gets close tothe natural frequency of bluff body the phenomenon calledldquosynchronizationrdquo or ldquolock-inrdquo occurs and the oscillatingamplitude can reach a high level Meanwhile for a bluff bodywith square-section under high Reynolds numbers whenwind speed exceeds a critical value the system begins toundergo aerodynamic instability and the phenomenon calledgalloping occurs To utilize the two phenomena for har-vesting flow energy some devices have been developed to

Hindawi Publishing CorporationJournal of SensorsVolume 2016 Article ID 2353517 12 pageshttpdxdoiorg10115520162353517

2 Journal of Sensors

Energy source Valve Vibratory body

Forward information

Feedback information

Figure 1 Feedback mechanism of self-excited vibration

produce energy from structural vibrations induced by windor water flow

Taylor et al [8] andAllen and Smits [9] described an ldquoeelrdquoequipment and investigated the utilization of flexible andflag-like PVDF (polyvinylidene fluoride) sheets to harvest powerin a water channel Robbins et al [10] investigated a similarmethod for utilizing PVDF sheet to generate power in a windtunnel

At Marine Renewable Energy Laboratory (MRELab) ofUniversity of Michigan Raghavan and Bernitsas [11] and Leeand Bernitsas [12] have developed an ocean energy harvest-ing system called VIVACE (Vortex-Induced Vibration forAquatic Clean Energy) The device harvests energy from theoscillations induced by vortex shedding from circular cylin-ders The factors such as mass ratio mechanical dampingReynolds numbers and aspect ratio (length to diameter) ofthe cylinder were investigated Wu et al [13] provided a CFDcode to solve VIV problem of a single cylinder with PTC(passive turbulence control) In addition a CFD code hasbeen developed by Ding et al [14] in OpenFOAM to solveVIV of multiple circular cylinders

At University of Virginia Mehmood et al [15] andAbdelkefi et al [16] developed a method for extracting flowenergy by using a wake oscillator model and direct numericalsimulation method It was reported that the presence of thehardening behavior was due to aerodynamic nonlinearityThey also explained the decisive effects of the load resis-tance on the synchronization region and level of the powerharvesting The maximum level of the power harvestingcould appear simultaneously with the minimum value inthe transverse displacement due to a shunt damping effectBesides Abdelkefi and his coauthors [17 18] investigated thepotential utility of harvesting energy from base excitationswith VIV or galloping Barrero-Gil et al [19] theoreticallyinvestigated the feasibility of energy harvesting from gallop-ing structures They used a cubic polynomial to representthe aerodynamic characteristics and obtained an expressionfor the harnessing of the flow energy To investigate thepossibility of harvesting energy from other flow-inducedvibrations various researches have been performed [20ndash22]For extracting energy from flutter phenomenon Abdelkefiand Nuhait [20] discussed the effects of a cambered wing-based piezoaeroelastic energy harvester Both linear andnonlinear mathematical models have been developed topredict the performance of the device

It is worth noting that both vortex-induced vibration andgalloping of a square-section structure can be utilized inthe same piezoelectric wind energy harvesting equipmentbecause both of the two phenomena can cause large oscil-lating amplitude that can be good power source to excite

Table 1 Properties of electromechanical system

Symbol Property Value119872 Mass of the cylinder per unit length 044 kg119870 Stiffness per unit length 854Nm119862 Damping per unit length 192Nms119863 Side length of the cylinder 15 cm120585 Damping ratio 00013119891119899

Natural frequency 7006Hz119888119901

Capacitance 120 nF120579 Electromechanical coupling 155119864 minus 3NV

the piezoelectric device to vibrate However there are fewliteratures focusing on considering of the two phenomena inthe same energy harvesting system

This paper describes a piezoelectric device consideringboth vortex-induced vibration and galloping of a square-section cylinder for harvesting wind energy In present worka two-dimensional dynamical system is used to investigatethe coupling of outflow field cylinder and the piezoelectrictransducer The parameters in current numerical calculationsystem are listed in Table 1

The aeroelectromechanical coupled governing equationsare solved to determine the level of harvested energy fromSEV The harvested power is investigated under differentoperating conditions including the reduced wind velocitiesand load resistance In Section 2 the physical and mathe-matical models are discussed and validation of the fluid flowsolver and the coupling scheme with previous experimentaldata is presented In Section 3 the open-circuit system withinfinite load resistance is discussed In this part a simplifiedmodel is introduced to calculate the open-circuit voltageoutput In Section 4 the close-circuit system with differentload resistance ranging from 100Ω to 11198646Ω is discussed theeffects of reduced velocity and load resistance on the oscil-lation amplitudes fluctuating lift coefficient voltage outputandharvested power are studied In Section 5 the conclusionsare presented

2 Physical and Mathematical Models

21 Physical Model of SEV A simplified SEV-based energyharvester configuration was explored The model consists ofa spring-mounted square-section cylinder and a piezoelectrictransducer attached to its transverse degree of freedom Thecylinder vibrates freely under the action of an incoming flowin the transverse direction which is shown in Figure 2 Thesystem has a mass per unit length 119872 a spring-coefficient119870 damping 119862 and an electric resistance 119877 The equationgoverning the mechanical system is

119872( + 2120585120596119899 + 120596119899

2

119910) = 119865119910=1

21205881198802

119863119862119910 (1)

where119865119910is the fluid force per unit length in the normal direc-

tion to the incoming flow the dot symbol is differentiationwith respect to time 119905 120588 is the density of the incoming flow119880 is the velocity of the incident flow 119863 is the characteristicdimension of the body normal to the flow and 119862

119910is

Journal of Sensors 3

U

(a)

R

U

PZT

(b)

Figure 2 (a)The experimental setup of the vibrational system (b) Schematic of the proposed square-cylinder based piezoaeroelastic energyharvester

(a)

Square columnInlet Outlet

Top

Bottom

y

x

(b)

Figure 3 (a) The computational grid in present work (b) The boundary conditions of the system

the instantaneous fluid force coefficient in the transversedirection to the incident flow For galloping particularly acubic polynomial can be used to approximate the verticalfluid force coefficient 119862

119910[23]

119862119910= [1198861

119880+ 1198863(

119880)

3

] (2)

where 1198861and 1198863are empirical coefficients measured in static

tests It is noted that if the cross-section of the structure issymmetric about a line in the flow direction through thecenter of the section only odd harmonics 119886

1 1198863 and so forth

in the series are nonzero [7]

The relationship between the parameters of the 119872-119862-119870system can be presented as

119870 = 120596119899

2

sdot 119872

119862 = 2120585 sdot 119872 sdot 120596119899

(3)

where 120596119899is the natural frequency of oscillations and 120585 is the

damping ratioFor the axial symmetry of the system a two-dimensional

model was used to describe the energy harvesting systemwhich was showed in Figure 2 Figure 2(a) presents the realvibrational system The structured grid is used for its com-putational stability to solve the moving grid problem and thenearwall grid is shown in Figure 3(a)The cylinder undergoes


SEV when the harvester is subjected to an incoming flowFigure 3(b) shows the boundary condition of the systemIf the free stream velocity comes to a critical value of theonset of ldquosynchronizationrdquo the vortex shedding frequencycan be locked into the cylinder oscillating frequency theamplitude can becomepretty high If the flowvelocity exceedsthe second critical value galloping occurs in the transversedirection and higher level amplitude can be obtained

22 Electromechanical Coupling Model The fluid-electrome-chanical coupling procedure for the 119872-119862-119870 system wasbriefly introduced by deriving the mathematical model of thesystem The continuity and Navier-Stokes (N-S) equationsand the one-dimensional constitutive equations of the piezo-electric transducer are given by

119863120588

119863119905+ 120588119878119896119896

= 0 (4)

120588119863119881119894

119863119905= minus

120597119901

120597119909119894

+

120597120591119894119895

120597119909119895

(5)

120588119863

119863119905= minus

120597

120597119905+119881119896120597119909

120597119909119896

(6)

Equation (6) is the total derivative operator 119901 is thepressure 120588 is the fluid density 119881

119894is fluid velocity vector 120591

119894119895

is the stress tensor and 119878119896119896is the strain rate tensor A nonslip

boundary condition on the surface of a moving body is givenas

119881 (119909 119910) = 119881Ω(119909 119910) 119909 119910 isin Ω (7)

where Ω is the wet surface of the deforming structure 119881Ωis

moving velocity of Ω during the fluid-structure interactionprocess and the cylinder moves up and down in the flowfield under the boundary conditions while the pressure 119875

Ωis

acting onΩ and the forcing term119865119910in the structural equation

shown in the following is generatedIn present work we focused on the effect of load resis-

tance 119877 on the energy harvesting system Two differentsystems open-circuit system and close-circuit system wereidentified through the value of investigated load resistance 119877

221 Electromechanical Coupling Model of the Open-CircuitSystem For the single-degree-of-freedom (SDOF) systemwith a resistive load (119877) (1) can be rewritten with an addi-tional electromechanical term and given by

119872 + 119862 + 119870119910 minus 120579119881 = 119865119910 (8)

120579 + 119888119901 +

119881

119877= 0 (9)

Or

120579 + 119888119901 = minus

119881

119877= 119868 (10)

For open-circuit condition the equations of the SDOFcoupling system with a resistive load (119877) can be rewritten as

119872 + 119862 + 119870119910 minus 120579119881oc = 119865119910 (11)

where 119881oc is the open-circuit voltage and 119868 is the current Inthis case if we set 119877 = infin (10) can be available

119910 = minus

119888119901

120579119881oc (12)

Equation (12) can be added into (8) as

119872 + 119862 + (1198961015840

+1205792

119888119901

)119910 = 119865119910 (13)

The parameter 1198961015840 = 120596119899

2

119898+1205792

119888119901is the couplingmechan-

ical stiffness Substituting (3) into (13) and introducingdimensionless variables ℎ = 119910119863 and 120591 = 120596

119899119905 a new equation

form can be given as

ℎ + 2120585ℎ + (120596119899

2

+1205792

119888119901119898)ℎ =

1205881198802

2119898119862119910 (14)

Substituting (14) into (13) and introducing another two-dimensionless mass ratio119898

lowast

= 1198981205881198632 and reduced velocity

119880119903= 119880(120596

119899119863)

ℎ + 2120585ℎ + (120596119899

2

+1205792

119888119901119898lowast

)ℎ

=119880lowast2

2119898lowast(1198861

ℎ

119880119903

+ 1198863(

ℎ

119880119903

)

3

)

(15)

Equation (15) expresses the dimensionless form of elec-tromechanical coupling For low Reynolds numbers thecoefficients 119886

1and 1198863can be given as

1198861= minus27 + 0017Re

1198863= 10 minus 0096Re minus 0001Re2

(16)

In cases of Re gt 200 the linear and nonlinear coefficientscan be given by Parkinson and Smithrsquos experimental results[24 25] where 119886

1= 23 and 119886

3= minus18

222 Mathematical Model of Electromechanical Dampingand Onset of Galloping Load resistance makes the dampingand natural frequency of the electromechanical system bedifferent from that without the electromechanical circuitTheimpact of variations in both the frequency and damping onthe system can be evaluated by analyzing the linear part of theelectromechanical coupled equations Considering the firstterm of the aerodynamic force 119862

119910only the equation group

can be rewritten as the following formation of first-orderdifferential equations

1= 1198832

2= minus(2120585120596

119899minus1205881198802

1198631198861

2119898)1198832minus 120596119899

2

1198831+

120579

1198981198833

3= minus

1

1198771198881199011198833

minus120579

1198881199011198832

(17)

where1198831= 119910119883

2= and119883

3= 119881


Equation (17) can be expressed as a vector form

= 119866 (119877)119883 (18)

where

119883 = (

1198831

1198832

1198833

)

119866 (119877)119883

=(((

(

0 1 0

minus120596119899

2

minus(2120585120596119899minus1205881198802

1198631198861

2119898)

120579

119898

0 minus120579

119888119901

minus1

119877119888119901

)))

)

(19)

The matrix 119866(119877)119883 has three different eigenvalues 120582119894

119894 = 1 2 3 Barrero-Gil et al [19] indicated that the first twoeigenvalues are similar to those of a pure galloping problem inthe absence of the piezoelectric load and the third eigenvalueis the result of the effect of electromechanical coupling Itis noted that the first two eigenvalues are conjugates (120582

2=

1205821) The real and imaginary parts respectively represent the

damping coefficient and the global frequency of the couplingsystem One property of a matrix is that the product of theeigenvalues equals the value of the determinant and hereit can be presented as 120582

112058221205823

= minus120596119899

2

119877119888119901 The first two

eigenvalues are complex conjugates thus the value of 12058211205822

is always nonnegative thus 1205823is always real negative and

the stability of the trivial solutions depends only on the firsttwo eigenvalues If the real part of 120582

1is negative the system

is stable on the other hand the system is unstable if thevalue is positive As a result the critical speed 119880

119892at 1205821=

0 corresponds to the onset of instability or galloping Theelectromechanical coupling power output can be obtained

119875 (119905) = (

119890minus119905119877119888119901120579120596119884max119877 minus 120579120596119884max119877 (cos (120596119905 + 120593) + 120596 sin (120596119905 + 120593) 119888

119901119877)

1 + 12059621198772119888119901

2)

2

119877minus1

(20)

where 119875(119905) is the power output of the system 120596 is thevibrational angular frequency and 120593 is phase angle

223 The Validation of the Solver for Fluid-Structure Inter-action The fluid-structure interaction solver of OpenFOAMis verified by comparing the vibrational displacement of thesquare cylinder with that of Blevins [7] Two square cylinderswith a fixed length-width ratio of 2 and different dampingratios are investigated The reduced velocity is set to 40to 160 The computation of synchronization phenomenonconsidering the critical wind speed and the instable gallopingresponse is emphatically investigated The results are shownin Figure 4 119860119863 refers to the nondimensional vibrationalamplitude Blevins [7] showed that the onset of synchro-nization of the square cylinder occurred when 119880

119903varied in

the range of 50 to 55 and the onset of galloping 119880119903was

ranged between 110 and 115 The predicted critical velocitiesof synchronization and galloping here agree well with thework of Blevins

The analysis of energy harvesting in electromechanicalcoupling of the system consists of two parts open-circuit andclose-circuit which are identified by the load resistance Andfor the open- and close-circuit system the vortex-inducedvibration (VIV) and the galloping phenomenon are bothconsideredThe scope of reduced velocity119880

119903ranges from 30

to 200

3 Results of Open-Circuit System

31 Energy Harvesting from the ldquoLock-Inrdquo Phenomenon ofVIV in Open-Circuit System Under the low wind speed thesynchronization phenomenon of a square cylinder can be

obtained Galloping could not happen when Reynolds num-ber is lower than 159 [26] However the synchronization phe-nomenon can be effectively used to harvest energy Figure 5shows the computational results of vibrational amplitudes liftcoefficient and the open-circuit voltage under low Reynoldsnumbers in an open-loop circuit system (119877 = infin)

The phenomenon synchronization can be divided intothree branches presynchronization synchronization andpostsynchronization In presynchronization part the ampli-tude grows in a stable law As shown in Figures 5(a) 5(b)and 5(c) when vortex shedding frequency is getting close tothe natural frequency of the cylinder the phenomenon ldquoflaprdquocan be clearly captured During this period it is worthnoting that there were two major dominant frequenciesin this region However to make a difference with the curvesshown in Figures 5(g) 5(h) and 5(i) the dominant frequencyin current presynchronization branch is the vortex sheddingfrequency shown in Figures 5(d) 5(e) and 5(f) the self-excited vibration gets into the branch of lock-in In this phasethe predicted amplitude quickly gets up to the stable peakvalue of 048119863 and vortex shedding frequency is locked at thenatural frequency of square cylinder

It is worth noting that considerable oscillating energy inthis branch is available for energy harvesting process and theoutput open-circuit voltage can reach 58 VWhen119880

119903is equal

to 58 the synchronization phenomenon bifurcates and ahigh level of modulations is observed in the time historiesof transverse displacement and open-circuit output voltageoutput as shown in Figures 5(g) 5(h) and 5(i) The ldquoflaprdquophenomenon occurs again in the branch of postsynchro-nization The amplitude reduces regularly and the dominantfrequency switches to the natural frequency Modulations


03

02

01

00

Non

dim

ensio

nal a

mpl

itude

s (AD

) U

L

D

3 6 9 12 15

Ur = UfnD

Damping ratio 120585

037 Blevins (1990)076 Blevins (1990)

037 current work076 current work

Figure 4 Comparison of displacement of self-excited vibration inpresent work to previous work of Blevins under different dampingratio

also occur in the time history of lift coefficient as shown inFigure 5(h) From all the predicted results the time historiesof transverse displacement and output open-circuit voltageoutput have the same trend This is expected according tothe Gauss law the voltage is directly related to transversedisplacement Figures 6(a)ndash6(d) indicate that the vorticitycontours of this instance and two different locations of thecylinder are identified by the offset of 119910 = plusmn048119863 to its initialposition A ldquo2Srdquo mode vortex street is clearly shown in theregion behind the square cylinder

Variations of transverse displacement and voltage outputare shown in Figure 7 It can be seen that in the open-circuitsystem the curves of transverse displacement and voltageoutput have three branches presynchronization synchro-nization and postsynchronizationThe best operating condi-tion is at119880

119903= 56 and in current configuration which means

that the relative wind velocity is equal to 039ms under theparameters in present workTherefore the energy harvestingcan be achieved from the synchronization phenomenon evenat low reduced velocity in open-circuit system

As shown in Figure 7 the system becomes instable againwhen flow velocity is higher at the end of the vortex-inducedvibration branch and galloping occurs when flow velocityrises to the critical value of the instability The amplitudegets up to a higher level while the oscillating frequency isin a lower level From the figure it is shown that the vortex-induced vibration ends at 119880

119903= 64

Meanwhile the oscillating becomes intensive at119880119903= 94

Figure 8 reflects the time history of transverse displacementat 119880119903= 10 the curve seems to be not as smooth as the syn-

chronization and the vibrational amplitude is much largerIt can be concluded that the onset velocity of galloping is upwith the resistance of the system

32 Effects of the Electrical Load Resistance on the Onset ofGalloping Figure 9(a) shows the variations of the real andimaginary part of the eigenvalues 120582

119894of the linear system It

can be seen that the electromechanical damping coefficientremains low in the presence of low load resistance valuesParticularly the phenomenon resistive shunt damping effectmakes the coupled electromechanical damping close to amaximum value of 033 However the natural frequencyremains stable at 423 rads before 119877 = 181198644Ω whichis referred to short global frequency For a higher 119877 thenatural frequency rises up to 452 rads This value of thenatural frequency is referred to the open global frequencyThe curve shows that load resistance can impact the naturalfrequency and damping ratio of an electromechanical systemFigure 9(b) shows the effect of the electrical load resistance onthe onset of the galloping It can be seen that the onset of thegalloping initially remains unchanged at 119880

119903= 75 when 119877 is

100Ω and increases with the enhancement of load resistanceThe value gets to the peak value at 119877 = 81198644Ω and decreasesif the load resistance keeps increasing

4 Results of Close-Circuit System

41 Energy Harvested from VIV in Close-Circuit SystemNumerical simulations are performed by utilizing proposeddamping coefficient 119862 and global frequency 120596

119899 The compu-

tational results of transverse displacement and output voltageare shown in Figures 10(a)ndash10(d) It is clear that when 119880

119903

falls into the range of 5345 to 6359 the self-excited vibrationof the square cylinder spans presynchronization synchro-nization and postsynchronization Figure 10(a) representsthe energy harvesting system with the load resistance inthe aforementioned velocity rangeThe synchronization phe-nomenon begins at about the same velocity (119880

119903= 5345)

for all load resistances due to the fact that the global naturalfrequency roughly remains unchanged with the load resis-tance (in Figure 9(a)) It is worth noting that the amplitudeof transverse displacement significantly varies with the loadresistance The maximum value of oscillation amplitude is119910119863 = 0396 when 119877 = 100Ω and decreases to 0348 when119877 = 11198643Ω and to 0335 when 119877 = 11198644Ω For 119877 = 11198645Ω theamplitude comes to the minimum value due to the fact thatthe electromechanical coupling damping reaches maximumwhen 119877 = 11198645Ω as shown in Figure 9(a)

On the other hand Figure 10(a) shows that the widthof the synchronization region is approximately unchangedwith electromechanical coupling damping caused by loadresistance Variation trends of the root mean square close-circuit voltage output in Figures 10(b) and 10(c) are shownin two different forms Figure 10(b) shows the variation trendof voltage output by the reduced velocity under different loadresistanceThe voltage output gets higher with the increasingof the load resistance the maximum is attained when 119880

119903=

578 and 119877 = 11198646Ω When 119877 = 11198645Ω and 119880119903= 596

the maximum voltage output is about 0288V and when119877 = 1KΩ and 119880

119903= 605 the maximum decreases to

0023V When 119877 = 100Ω the voltage output is lower andthe maximum value is 181119864 minus 3V It can be concluded


006

003

000

minus003

minus006320 330 340 350 360 370

tT

yD

(a)

03

00

minus03

320 330 340 350 360 370

tT

CL

(b)03

00

minus03320 330 340 350 360 370

tT

Volta

ge (V

)

(c)

tT

yD

06

00

minus0625 26 27 28 29 30

(d)

tT

CL

06

00

minus0625 26 27 28 29 30

(e)

tT

Volta

ge (V

)

8

4

0

minus4

minus825 26 27 28 29 30

(f)

tT

yD

012

000

minus01230 40 50 60 70

(g)

tT

CL

04

00

minus0430 40 50 60 70

(h)

tT

Volta

ge (V

)

02

00

minus0230 40 50 60 70

(i)

Figure 5 Time histories of transverse displacement lift and output voltage for regime of presynchronization (a b c) synchronization (d ef) postsynchronization (g h i) in the open-circuit system

that the voltage output of the close-circuit system increaseswith the increasing of load resistance This conclusion canalso be obtained from Figure 10(c) which shows the varia-tions of voltage output with load resistance under differentflow velocities As the load resistance increases the voltageincreases in a stable law Figure 10(d) shows variations of

harvested power with flow velocity under different loadresistanceThe harvested power is computed with the voltageaccording to following equation

119875har =1198812

rms119877

(21)


(a)

Y = 048D

(b)

(c)

Y = 048D

(d)

Figure 6 Vorticity contours of the vortex street (a) the original position of the cylinder (b) the cylinder moving to the top position in theflow (c) the cylinder moving back to the original position (d) the cylinder moving to the bottom position in the flow

55 60 65 7000

01

02

03

04

05

AD

Transverse displacementOpen-circuit output voltage

0

2

4

6

8

Ur = UfnD

Ymax = 048D

Vmax = 58V

Vrm

s(v

olts)

Figure 7 Variation trends of the transverse displacement and out-put voltage of the vortex-induced vibration of open-circuit system

where 119881rms indicates the root mean square value or theeffective value of the voltage output and 119877 represents the loadresistance across the piezoelectric transducer Figure 10(d)shows that the maximum harvested power can be obtainedunder all considered 119877 in the synchronization system when119877 = 11198646Ω The harvested power reaches its maximum value119875har = 114119864 minus 5W A notable phenomenon is that for allconsidered119877 the harvested power can also be enhanced withthe increase of 119877

42 Energy Harvesting Resulted from Galloping in the Close-Circuit System To analyze the nonlinear problem of theelectromechanical coupling system the governing equa-tions are directed computed on OpenFOAM Following the

synchronization branch the galloping phenomenon occurswith higher flow velocities The transverse displacementcan reach a more high level than that of synchronizationbranch thus more energy can be harvested However theoscillating frequency of galloping is much lower than that ofsynchronization

Figure 11 shows the transverse displacement voltageoutput and harvested energy under different119877 caused by dif-ferent reduced velocities of galloping of the square cylinderThe load resistance varies with the onset speed of gallopingIn Figure 11(a) the transverse displacement rises with theincrease of free stream velocity The onset speed reaches itsmaximum value when 119877 = 21198645Ω the result also shows thatthe load resistance contributes to variations of the amplitudeof the transverse displacement The transverse displacementhas the smallest value when 119877 = 21198645Ω for the electrome-chanical damping is in the highest level However the voltageoutput could not be accompanied with an increase in the har-vested power as shown in Figure 11(b) It is worth noting thatthere is an optimum value of load resistance for maximizingthe harvested power The maximum level is obtained when119877 = 21198645Ω Under this selected load resistance the harvestedpower is nearly 14 watts with the reduced velocity of 200 orso

5 Conclusions

In present work energy harvesting fromboth vortex-inducedvibration and galloping phenomenon from SEV of a squarecylinder has been investigated under different wind speedsin platform of OpenFOAM The analysis of electromechan-ical coupling system is carried out under both open- andclose-circuit parts In open-circuit system three branches


50

25

00

minus25

minus50650 655 660 665 670 675 680

AD

tT

Figure 8 Time history of transverse displacement of galloping of the square cylinder at 119880119903= 100

00

01

02

03

Damping coefficient

Dam

ping

coeffi

cien

t

Natural frequency

42

43

44

45

(Hz)

103 104 105 106

R (ohms)

(a)

6

8

10

12

14

16

Onset of galloping velocity

Ur=

Uf

nD

102 103 104 105 106 107

R (ohms)

(b)

Figure 9 (a) Variations of damping coefficient and natural frequency of the linear systemwith the load resistance (b) Variations of gallopingvelocity with the load resistance

in vortex-induced vibration have been obtained and thevibrational displacement of both VIV and galloping has beeninvestigated It can be concluded that in VIV the vibrationaldisplacement curves were varying in a sinusoidal law whichis different from the galloping regime The vortex sheddingpattern has been shown by vortex contoursThemax value ofopen-circuit voltage output in VIV regime has been obtainedto 58V at 119880

119903= 58 in synchronization regime For close-

circuit system the effects of load resistance on the naturalfrequency and system damping have been analyzed theharvested power and transverse displacement of the squarecylinder are investigated For VIV the maximum harvestedpower reached maximum 119875har = 114119864 minus 5W when 119877 =

11198646Ω For galloping the harvested power is 14W when119877 = 21198645Ω It can be concluded that synchronization phe-nomenon can be utilized to harvest power when the velocity

of wind is low When wind speed gets higher to gallopingthe open-circuit voltage output increases as the wind speedincreases In close-circuit system the harvested power canbe enhanced by increasing load resistance in the low windspeed region For highwind speed applications the harvestedpower can be optimized to obtain theminimumof oscillatingdisplacement by choosing adaptive load resistance

Nomenclature

119880 Flow velocity (ms)119863 Characteristic length (m)119910 Transverse displacement (m)119872 Mass (Kg)119870 Stiffness (Nm)119862 Damping (Nms)


00

01

02

03

04y

maxD

R = 100Ω

54 56 58 60 62 64

Ur = UfnD

R = 1KΩ

R = 100 KΩ

R = 1MΩ

R = 10 KΩ

(a)

54 56 58 60 62 64

102

101

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

Ur = UfnD

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(b)

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

102 103 104 105 106

R (ohms)

Ur = 54

Ur = 56

Ur = 58

Ur = 60Ur = 62

(c)

5654 58 60 62 64

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

Ur = UfnD

P(w

att)

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(d)

Figure 10 Variations of (a) transverse displacement ((b) and (c)) voltage output (d) harvested power with the nondimensional flow velocity

119891 Frequency (Hz)119877 Load resistance (ohms)119865 Force (N)Re Reynolds number119875 Pressure (Pa)120596 Circular frequency (Hz)120585 Damping coefficient120579 Electromechanical coupling (NV)120591 Stress (N)Ω Deforming wet surface120582119894 Eigenvalues

119880119903 Reduced velocity

119891119899 Natural frequency

119875har Harnessed power119880rms Root mean square voltage

Competing Interests

All the funds do not lead to any competing interests regardingthe publication of this paper

Acknowledgments

This work was supported by Specialized Research Fund forthe Natural Science Foundation of Guangdong Province(Grant no 2015A030310182) Open Fund of Chongqing


Transverse displacement

R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

9 12 15 18

Ur = UfnD

6

4

2

0

ym

axD

(a)

9 12 15 180

10

20

30

40

50

60Voltage output

Ur = UfnD

V(V

)

R = 2e2Ω

R = 2e3ΩR = 2e4Ω

R = 2e5ΩR = 2e6Ω

(b)

00

04

08

12

Harvested power

R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

Ur = UfnD

P(W

)

9 12 15 18

(c)

Figure 11 Variations of (a) transverse displacement and (b) voltage output (c) harvested power with the nondimensional flow velocity

University Key Laboratory of Low-Grade Energy UtilizationTechnologies and Systems (Grant no LLEUTS-201610) andOpen Fund of Jiangsu Engineering Research Center onMeteorological Energy Using and ControlC-MEIC NanjingUniversity of Information Science amp Technology (Grant noKCMEIC02)

References

[1] I F Akyildiz W Su Y Sankarasubramaniam and E CayircildquoWireless sensor networks a surveyrdquo Computer Networks vol38 no 4 pp 393ndash422 2002

[2] C Alippi R Camplani C Galperti and M Roveri ldquoA robustadaptive solar-powered WSN framework for aquatic environ-mental monitoringrdquo IEEE Sensors Journal vol 11 no 1 pp 45ndash55 2011

[3] T Arampatzis J Lygeros and S Manesis ldquoA survey of appli-cations of wireless sensors and wireless sensor networksrdquo inProceedings of the 20th IEEE International Symposium on Intel-ligent Control (ISIC rsquo05) and the13th Mediterranean Conferenceon Control and Automation (MED rsquo05) pp 719ndash724 LimassolCyprus June 2005

[4] D Brunelli C Moser L Thiele and L Benini ldquoDesign ofa solar-harvesting circuit for batteryless embedded systemsrdquo


IEEE Transactions on Circuits and Systems I Regular Papers vol56 no 11 pp 2519ndash2528 2009

[5] W K G Seah Z A Eu and H-P Tan ldquoWireless sensor net-works powered by ambient energy harvesting (WSN-HEAP)mdashsurvey and challengesrdquo in Proceedings of the 1st InternationalConference on Wireless Communication Vehicular TechnologyInformation Theory and Aerospace amp Electronic Systems Tech-nology (Wireless VITAE rsquo09) pp 1ndash5 Aalborg Denmark May2009

[6] G Werner-Allen K Lorincz M Welsh et al ldquoDeploying awireless sensor network on an active volcanordquo IEEE InternetComputing vol 10 no 2 pp 18ndash25 2006

[7] R D Blevins ldquoFlow-induced vibrationrdquo 1990[8] G W Taylor J R Burns S M Kammann W B Powers and

T R Welsh ldquoThe energy harvesting Eel a small subsurfaceoceanriver power generatorrdquo IEEE Journal of Oceanic Engineer-ing vol 26 no 4 pp 539ndash547 2001

[9] J J Allen and A J Smits ldquoEnergy harvesting EELrdquo Journal ofFluids and Structures vol 15 no 3-4 pp 629ndash640 2001

[10] W P Robbins I Marusic D Morris and T O Novak ldquoWind-generated electrical energy using flexible piezoelectric ma-teialsrdquo in Proceedings of the ASME International MechanicalEngineering Congress and Exposition pp 581ndash590 Chicago IllUSA November 2006

[11] K Raghavan and M M Bernitsas ldquoExperimental investigationof Reynolds number effect on vortex induced vibration of rigidcircular cylinder on elastic supportsrdquo Ocean Engineering vol38 no 5-6 pp 719ndash731 2011

[12] J H Lee and M M Bernitsas ldquoHigh-damping high-ReynoldsVIV tests for energy harnessing using the VIVACE converterrdquoOcean Engineering vol 38 no 16 pp 1697ndash1712 2011

[13] W Wu M M Bernitsas and K Maki ldquoRANS simulation vsexperiments of flow induced motion of circular cylinder withpassive turbulence control at 35 000 lt Re lt 130 000rdquo inProceedings of the ASME 2011 30th International Conference onOcean Offshore and Arctic Engineering pp 733ndash744 AmericanSociety ofMechanical Engineers Rotterdam Netherlands June2011

[14] L DingMM Bernitsas and E S Kim ldquo2-DURANS vs exper-iments of flow inducedmotions of two circular cylinders in tan-dem with passive turbulence control for 30000ltRelt105000rdquoOcean Engineering vol 72 pp 429ndash440 2013

[15] A Mehmood A Abdelkefi M R Hajj A H Nayfeh I AkhtarandA O Nuhait ldquoPiezoelectric energy harvesting from vortex-induced vibrations of circular cylinderrdquo Journal of Sound andVibration vol 332 no 19 pp 4656ndash4667 2013

[16] A Abdelkefi M R Hajj and A H Nayfeh ldquoPhenomena andmodeling of piezoelectric energy harvesting from freely oscil-lating cylindersrdquo Nonlinear Dynamics vol 70 no 2 pp 1377ndash1388 2012

[17] A Abdelkefi M R Hajj and A H Nayfeh ldquoPiezoelectricenergy harvesting from transverse galloping of bluff bodiesrdquoSmart Materials and Structures vol 22 no 1 Article ID 0150142013

[18] A Abdelkefi M R Hajj and A H Nayfeh ldquoPower harvest-ing from transverse galloping of square cylinderrdquo NonlinearDynamics vol 70 no 2 pp 1355ndash1363 2012

[19] A Barrero-Gil G Alonso and A Sanz-Andres ldquoEnergy har-vesting from transverse gallopingrdquo Journal of Sound and Vibra-tion vol 329 no 14 pp 2873ndash2883 2010

[20] A Abdelkefi and A O Nuhait ldquoModeling and performanceanalysis of cambered wing-based piezoaeroelastic energy har-vestersrdquo Smart Materials and Structures vol 22 no 9 ArticleID 095029 2013

[21] H L Dai A Abdelkefi and LWang ldquoPiezoelectric energy har-vesting from concurrent vortex-induced vibrations and baseexcitationsrdquo Nonlinear Dynamics vol 77 no 3 pp 967ndash9812014

[22] Z Yan and A Abdelkefi ldquoNonlinear characterization of con-current energy harvesting from galloping and base excitationsrdquoNonlinear Dynamics vol 77 no 4 pp 1171ndash1189 2014

[23] J P den Hartog Mechanical Vibrations Dover Mineola NYUSA 1956

[24] G Parkinson ldquoMathematical models of flow-induced vibra-tions of bluff bodiesrdquo in Flow-Induced Structural Vibrations pp81ndash127 Springer Berlin Germany 1974

[25] G V Parkinson and J D Smith ldquoThe square prism as an aeroe-lastic non-linear oscillatorrdquoThe Quarterly Journal of Mechanicsand Applied Mathematics vol 17 no 2 pp 225ndash239 1964

[26] A Barrero-Gil A Sanz-Andres and G Alonso ldquoHysteresis intransverse galloping the role of the inflection pointsrdquo Journal ofFluids and Structures vol 25 no 6 pp 1007ndash1020 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Energy source Valve Vibratory body

Forward information

Feedback information

Figure 1 Feedback mechanism of self-excited vibration

produce energy from structural vibrations induced by windor water flow

Taylor et al [8] andAllen and Smits [9] described an ldquoeelrdquoequipment and investigated the utilization of flexible andflag-like PVDF (polyvinylidene fluoride) sheets to harvest powerin a water channel Robbins et al [10] investigated a similarmethod for utilizing PVDF sheet to generate power in a windtunnel

At Marine Renewable Energy Laboratory (MRELab) ofUniversity of Michigan Raghavan and Bernitsas [11] and Leeand Bernitsas [12] have developed an ocean energy harvest-ing system called VIVACE (Vortex-Induced Vibration forAquatic Clean Energy) The device harvests energy from theoscillations induced by vortex shedding from circular cylin-ders The factors such as mass ratio mechanical dampingReynolds numbers and aspect ratio (length to diameter) ofthe cylinder were investigated Wu et al [13] provided a CFDcode to solve VIV problem of a single cylinder with PTC(passive turbulence control) In addition a CFD code hasbeen developed by Ding et al [14] in OpenFOAM to solveVIV of multiple circular cylinders

At University of Virginia Mehmood et al [15] andAbdelkefi et al [16] developed a method for extracting flowenergy by using a wake oscillator model and direct numericalsimulation method It was reported that the presence of thehardening behavior was due to aerodynamic nonlinearityThey also explained the decisive effects of the load resis-tance on the synchronization region and level of the powerharvesting The maximum level of the power harvestingcould appear simultaneously with the minimum value inthe transverse displacement due to a shunt damping effectBesides Abdelkefi and his coauthors [17 18] investigated thepotential utility of harvesting energy from base excitationswith VIV or galloping Barrero-Gil et al [19] theoreticallyinvestigated the feasibility of energy harvesting from gallop-ing structures They used a cubic polynomial to representthe aerodynamic characteristics and obtained an expressionfor the harnessing of the flow energy To investigate thepossibility of harvesting energy from other flow-inducedvibrations various researches have been performed [20ndash22]For extracting energy from flutter phenomenon Abdelkefiand Nuhait [20] discussed the effects of a cambered wing-based piezoaeroelastic energy harvester Both linear andnonlinear mathematical models have been developed topredict the performance of the device

It is worth noting that both vortex-induced vibration andgalloping of a square-section structure can be utilized inthe same piezoelectric wind energy harvesting equipmentbecause both of the two phenomena can cause large oscil-lating amplitude that can be good power source to excite

Table 1 Properties of electromechanical system

Symbol Property Value119872 Mass of the cylinder per unit length 044 kg119870 Stiffness per unit length 854Nm119862 Damping per unit length 192Nms119863 Side length of the cylinder 15 cm120585 Damping ratio 00013119891119899

Natural frequency 7006Hz119888119901

Capacitance 120 nF120579 Electromechanical coupling 155119864 minus 3NV

the piezoelectric device to vibrate However there are fewliteratures focusing on considering of the two phenomena inthe same energy harvesting system

This paper describes a piezoelectric device consideringboth vortex-induced vibration and galloping of a square-section cylinder for harvesting wind energy In present worka two-dimensional dynamical system is used to investigatethe coupling of outflow field cylinder and the piezoelectrictransducer The parameters in current numerical calculationsystem are listed in Table 1

The aeroelectromechanical coupled governing equationsare solved to determine the level of harvested energy fromSEV The harvested power is investigated under differentoperating conditions including the reduced wind velocitiesand load resistance In Section 2 the physical and mathe-matical models are discussed and validation of the fluid flowsolver and the coupling scheme with previous experimentaldata is presented In Section 3 the open-circuit system withinfinite load resistance is discussed In this part a simplifiedmodel is introduced to calculate the open-circuit voltageoutput In Section 4 the close-circuit system with differentload resistance ranging from 100Ω to 11198646Ω is discussed theeffects of reduced velocity and load resistance on the oscil-lation amplitudes fluctuating lift coefficient voltage outputandharvested power are studied In Section 5 the conclusionsare presented

2 Physical and Mathematical Models

21 Physical Model of SEV A simplified SEV-based energyharvester configuration was explored The model consists ofa spring-mounted square-section cylinder and a piezoelectrictransducer attached to its transverse degree of freedom Thecylinder vibrates freely under the action of an incoming flowin the transverse direction which is shown in Figure 2 Thesystem has a mass per unit length 119872 a spring-coefficient119870 damping 119862 and an electric resistance 119877 The equationgoverning the mechanical system is

119872( + 2120585120596119899 + 120596119899

2

119910) = 119865119910=1

21205881198802

119863119862119910 (1)

where119865119910is the fluid force per unit length in the normal direc-

tion to the incoming flow the dot symbol is differentiationwith respect to time 119905 120588 is the density of the incoming flow119880 is the velocity of the incident flow 119863 is the characteristicdimension of the body normal to the flow and 119862

119910is


U

(a)

R

U

PZT

(b)


(a)


Top

Bottom

y

x

(b)



119910[23]

119862119910= [1198861

119880+ 1198863(

119880)

3

] (2)






119870 = 120596119899

2

sdot 119872

119862 = 2120585 sdot 119872 sdot 120596119899

(3)







119863120588

119863119905+ 120588119878119896119896

= 0 (4)

120588119863119881119894

119863119905= minus

120597119901

120597119909119894

+

120597120591119894119895

120597119909119895

(5)

120588119863

119863119905= minus

120597

120597119905+119881119896120597119909

120597119909119896

(6)



119894119895



119881 (119909 119910) = 119881Ω(119909 119910) 119909 119910 isin Ω (7)



Ωis





119872 + 119862 + 119870119910 minus 120579119881 = 119865119910 (8)

120579 + 119888119901 +

119881

119877= 0 (9)

Or

120579 + 119888119901 = minus

119881

119877= 119868 (10)


119872 + 119862 + 119870119910 minus 120579119881oc = 119865119910 (11)


119910 = minus

119888119901

120579119881oc (12)


119872 + 119862 + (1198961015840

+1205792

119888119901

)119910 = 119865119910 (13)

The parameter 1198961015840 = 120596119899

2

119898+1205792





ℎ + 2120585ℎ + (120596119899

2

+1205792

119888119901119898)ℎ =

1205881198802

2119898119862119910 (14)


lowast


119880119903= 119880(120596

119899119863)

ℎ + 2120585ℎ + (120596119899

2

+1205792

119888119901119898lowast

)ℎ

=119880lowast2

2119898lowast(1198861

ℎ

119880119903

+ 1198863(

ℎ

119880119903

)

3

)

(15)



1198861= minus27 + 0017Re

1198863= 10 minus 0096Re minus 0001Re2

(16)


1= 23 and 119886

3= minus18




1= 1198832

2= minus(2120585120596

119899minus1205881198802

1198631198861

2119898)1198832minus 120596119899

2

1198831+

120579

1198981198833

3= minus

1

1198771198881199011198833

minus120579

1198881199011198832

(17)

where1198831= 119910119883

2= and119883

3= 119881



= 119866 (119877)119883 (18)

where

119883 = (

1198831

1198832

1198833

)

119866 (119877)119883

=(((

(

0 1 0

minus120596119899

2

minus(2120585120596119899minus1205881198802

1198631198861

2119898)

120579

119898

0 minus120579

119888119901

minus1

119877119888119901

)))

)

(19)



2=



112058221205823

= minus120596119899

2

119877119888119901 The first two






119892at 1205821=


119875 (119905) = (


119901119877)

1 + 12059621198772119888119901

2)

2

119877minus1

(20)



119903varied in





to 200






119903is equal



03

02

01

00

Non

dim

ensio

nal a

mpl

itude

s (AD

) U

L

D

3 6 9 12 15

Ur = UfnD


037 Blevins (1990)076 Blevins (1990)








119903= 64







119903= 75 when 119877 is




119899 The compu-


119903


119903= 5345)



119903=

578 and 119877 = 11198646Ω When 119877 = 11198645Ω and 119880119903= 596





006

003

000

minus003

minus006320 330 340 350 360 370

tT

yD

(a)

03

00

minus03

320 330 340 350 360 370

tT

CL

(b)03

00

minus03320 330 340 350 360 370

tT

Volta

ge (V

)

(c)

tT

yD

06

00

minus0625 26 27 28 29 30

(d)

tT

CL

06

00

minus0625 26 27 28 29 30

(e)

tT

Volta

ge (V

)

8

4

0

minus4

minus825 26 27 28 29 30

(f)

tT

yD

012

000

minus01230 40 50 60 70

(g)

tT

CL

04

00

minus0430 40 50 60 70

(h)

tT

Volta

ge (V

)

02

00

minus0230 40 50 60 70

(i)




119875har =1198812

rms119877

(21)


(a)

Y = 048D

(b)

(c)

Y = 048D

(d)


55 60 65 7000

01

02

03

04

05

AD


0

2

4

6

8

Ur = UfnD

Ymax = 048D

Vmax = 58V

Vrm

s(v

olts)






5 Conclusions



50

25

00

minus25

minus50650 655 660 665 670 675 680

AD

tT


00

01

02

03

Damping coefficient

Dam

ping

coeffi

cien

t

Natural frequency

42

43

44

45

(Hz)

103 104 105 106

R (ohms)

(a)

6

8

10

12

14

16


Ur=

Uf

nD

102 103 104 105 106 107

R (ohms)

(b)







Nomenclature



00

01

02

03

04y

maxD

R = 100Ω

54 56 58 60 62 64

Ur = UfnD

R = 1KΩ

R = 100 KΩ

R = 1MΩ

R = 10 KΩ

(a)

54 56 58 60 62 64

102

101

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

Ur = UfnD

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(b)

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

102 103 104 105 106

R (ohms)

Ur = 54

Ur = 56

Ur = 58

Ur = 60Ur = 62

(c)

5654 58 60 62 64

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

Ur = UfnD

P(w

att)

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(d)






Competing Interests


Acknowledgments




R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

9 12 15 18

Ur = UfnD

6

4

2

0

ym

axD

(a)

9 12 15 180

10

20

30

40

50

60Voltage output

Ur = UfnD

V(V

)

R = 2e2Ω

R = 2e3ΩR = 2e4Ω

R = 2e5ΩR = 2e6Ω

(b)

00

04

08

12

Harvested power

R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

Ur = UfnD

P(W

)

9 12 15 18

(c)



References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










U

(a)

R

U

PZT

(b)


(a)


Top

Bottom

y

x

(b)



119910[23]

119862119910= [1198861

119880+ 1198863(

119880)

3

] (2)






119870 = 120596119899

2

sdot 119872

119862 = 2120585 sdot 119872 sdot 120596119899

(3)







119863120588

119863119905+ 120588119878119896119896

= 0 (4)

120588119863119881119894

119863119905= minus

120597119901

120597119909119894

+

120597120591119894119895

120597119909119895

(5)

120588119863

119863119905= minus

120597

120597119905+119881119896120597119909

120597119909119896

(6)



119894119895



119881 (119909 119910) = 119881Ω(119909 119910) 119909 119910 isin Ω (7)



Ωis





119872 + 119862 + 119870119910 minus 120579119881 = 119865119910 (8)

120579 + 119888119901 +

119881

119877= 0 (9)

Or

120579 + 119888119901 = minus

119881

119877= 119868 (10)


119872 + 119862 + 119870119910 minus 120579119881oc = 119865119910 (11)


119910 = minus

119888119901

120579119881oc (12)


119872 + 119862 + (1198961015840

+1205792

119888119901

)119910 = 119865119910 (13)

The parameter 1198961015840 = 120596119899

2

119898+1205792





ℎ + 2120585ℎ + (120596119899

2

+1205792

119888119901119898)ℎ =

1205881198802

2119898119862119910 (14)


lowast


119880119903= 119880(120596

119899119863)

ℎ + 2120585ℎ + (120596119899

2

+1205792

119888119901119898lowast

)ℎ

=119880lowast2

2119898lowast(1198861

ℎ

119880119903

+ 1198863(

ℎ

119880119903

)

3

)

(15)



1198861= minus27 + 0017Re

1198863= 10 minus 0096Re minus 0001Re2

(16)


1= 23 and 119886

3= minus18




1= 1198832

2= minus(2120585120596

119899minus1205881198802

1198631198861

2119898)1198832minus 120596119899

2

1198831+

120579

1198981198833

3= minus

1

1198771198881199011198833

minus120579

1198881199011198832

(17)

where1198831= 119910119883

2= and119883

3= 119881



= 119866 (119877)119883 (18)

where

119883 = (

1198831

1198832

1198833

)

119866 (119877)119883

=(((

(

0 1 0

minus120596119899

2

minus(2120585120596119899minus1205881198802

1198631198861

2119898)

120579

119898

0 minus120579

119888119901

minus1

119877119888119901

)))

)

(19)



2=



112058221205823

= minus120596119899

2

119877119888119901 The first two






119892at 1205821=


119875 (119905) = (


119901119877)

1 + 12059621198772119888119901

2)

2

119877minus1

(20)



119903varied in





to 200






119903is equal



03

02

01

00

Non

dim

ensio

nal a

mpl

itude

s (AD

) U

L

D

3 6 9 12 15

Ur = UfnD


037 Blevins (1990)076 Blevins (1990)








119903= 64







119903= 75 when 119877 is




119899 The compu-


119903


119903= 5345)



119903=

578 and 119877 = 11198646Ω When 119877 = 11198645Ω and 119880119903= 596





006

003

000

minus003

minus006320 330 340 350 360 370

tT

yD

(a)

03

00

minus03

320 330 340 350 360 370

tT

CL

(b)03

00

minus03320 330 340 350 360 370

tT

Volta

ge (V

)

(c)

tT

yD

06

00

minus0625 26 27 28 29 30

(d)

tT

CL

06

00

minus0625 26 27 28 29 30

(e)

tT

Volta

ge (V

)

8

4

0

minus4

minus825 26 27 28 29 30

(f)

tT

yD

012

000

minus01230 40 50 60 70

(g)

tT

CL

04

00

minus0430 40 50 60 70

(h)

tT

Volta

ge (V

)

02

00

minus0230 40 50 60 70

(i)




119875har =1198812

rms119877

(21)


(a)

Y = 048D

(b)

(c)

Y = 048D

(d)


55 60 65 7000

01

02

03

04

05

AD


0

2

4

6

8

Ur = UfnD

Ymax = 048D

Vmax = 58V

Vrm

s(v

olts)






5 Conclusions



50

25

00

minus25

minus50650 655 660 665 670 675 680

AD

tT


00

01

02

03

Damping coefficient

Dam

ping

coeffi

cien

t

Natural frequency

42

43

44

45

(Hz)

103 104 105 106

R (ohms)

(a)

6

8

10

12

14

16


Ur=

Uf

nD

102 103 104 105 106 107

R (ohms)

(b)







Nomenclature



00

01

02

03

04y

maxD

R = 100Ω

54 56 58 60 62 64

Ur = UfnD

R = 1KΩ

R = 100 KΩ

R = 1MΩ

R = 10 KΩ

(a)

54 56 58 60 62 64

102

101

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

Ur = UfnD

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(b)

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

102 103 104 105 106

R (ohms)

Ur = 54

Ur = 56

Ur = 58

Ur = 60Ur = 62

(c)

5654 58 60 62 64

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

Ur = UfnD

P(w

att)

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(d)






Competing Interests


Acknowledgments




R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

9 12 15 18

Ur = UfnD

6

4

2

0

ym

axD

(a)

9 12 15 180

10

20

30

40

50

60Voltage output

Ur = UfnD

V(V

)

R = 2e2Ω

R = 2e3ΩR = 2e4Ω

R = 2e5ΩR = 2e6Ω

(b)

00

04

08

12

Harvested power

R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

Ur = UfnD

P(W

)

9 12 15 18

(c)



References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119863120588

119863119905+ 120588119878119896119896

= 0 (4)

120588119863119881119894

119863119905= minus

120597119901

120597119909119894

+

120597120591119894119895

120597119909119895

(5)

120588119863

119863119905= minus

120597

120597119905+119881119896120597119909

120597119909119896

(6)



119894119895



119881 (119909 119910) = 119881Ω(119909 119910) 119909 119910 isin Ω (7)



Ωis





119872 + 119862 + 119870119910 minus 120579119881 = 119865119910 (8)

120579 + 119888119901 +

119881

119877= 0 (9)

Or

120579 + 119888119901 = minus

119881

119877= 119868 (10)


119872 + 119862 + 119870119910 minus 120579119881oc = 119865119910 (11)


119910 = minus

119888119901

120579119881oc (12)


119872 + 119862 + (1198961015840

+1205792

119888119901

)119910 = 119865119910 (13)

The parameter 1198961015840 = 120596119899

2

119898+1205792





ℎ + 2120585ℎ + (120596119899

2

+1205792

119888119901119898)ℎ =

1205881198802

2119898119862119910 (14)


lowast


119880119903= 119880(120596

119899119863)

ℎ + 2120585ℎ + (120596119899

2

+1205792

119888119901119898lowast

)ℎ

=119880lowast2

2119898lowast(1198861

ℎ

119880119903

+ 1198863(

ℎ

119880119903

)

3

)

(15)



1198861= minus27 + 0017Re

1198863= 10 minus 0096Re minus 0001Re2

(16)


1= 23 and 119886

3= minus18




1= 1198832

2= minus(2120585120596

119899minus1205881198802

1198631198861

2119898)1198832minus 120596119899

2

1198831+

120579

1198981198833

3= minus

1

1198771198881199011198833

minus120579

1198881199011198832

(17)

where1198831= 119910119883

2= and119883

3= 119881



= 119866 (119877)119883 (18)

where

119883 = (

1198831

1198832

1198833

)

119866 (119877)119883

=(((

(

0 1 0

minus120596119899

2

minus(2120585120596119899minus1205881198802

1198631198861

2119898)

120579

119898

0 minus120579

119888119901

minus1

119877119888119901

)))

)

(19)



2=



112058221205823

= minus120596119899

2

119877119888119901 The first two






119892at 1205821=


119875 (119905) = (


119901119877)

1 + 12059621198772119888119901

2)

2

119877minus1

(20)



119903varied in





to 200






119903is equal



03

02

01

00

Non

dim

ensio

nal a

mpl

itude

s (AD

) U

L

D

3 6 9 12 15

Ur = UfnD


037 Blevins (1990)076 Blevins (1990)








119903= 64







119903= 75 when 119877 is




119899 The compu-


119903


119903= 5345)



119903=

578 and 119877 = 11198646Ω When 119877 = 11198645Ω and 119880119903= 596





006

003

000

minus003

minus006320 330 340 350 360 370

tT

yD

(a)

03

00

minus03

320 330 340 350 360 370

tT

CL

(b)03

00

minus03320 330 340 350 360 370

tT

Volta

ge (V

)

(c)

tT

yD

06

00

minus0625 26 27 28 29 30

(d)

tT

CL

06

00

minus0625 26 27 28 29 30

(e)

tT

Volta

ge (V

)

8

4

0

minus4

minus825 26 27 28 29 30

(f)

tT

yD

012

000

minus01230 40 50 60 70

(g)

tT

CL

04

00

minus0430 40 50 60 70

(h)

tT

Volta

ge (V

)

02

00

minus0230 40 50 60 70

(i)




119875har =1198812

rms119877

(21)


(a)

Y = 048D

(b)

(c)

Y = 048D

(d)


55 60 65 7000

01

02

03

04

05

AD


0

2

4

6

8

Ur = UfnD

Ymax = 048D

Vmax = 58V

Vrm

s(v

olts)






5 Conclusions



50

25

00

minus25

minus50650 655 660 665 670 675 680

AD

tT


00

01

02

03

Damping coefficient

Dam

ping

coeffi

cien

t

Natural frequency

42

43

44

45

(Hz)

103 104 105 106

R (ohms)

(a)

6

8

10

12

14

16


Ur=

Uf

nD

102 103 104 105 106 107

R (ohms)

(b)







Nomenclature



00

01

02

03

04y

maxD

R = 100Ω

54 56 58 60 62 64

Ur = UfnD

R = 1KΩ

R = 100 KΩ

R = 1MΩ

R = 10 KΩ

(a)

54 56 58 60 62 64

102

101

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

Ur = UfnD

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(b)

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

102 103 104 105 106

R (ohms)

Ur = 54

Ur = 56

Ur = 58

Ur = 60Ur = 62

(c)

5654 58 60 62 64

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

Ur = UfnD

P(w

att)

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(d)






Competing Interests


Acknowledgments




R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

9 12 15 18

Ur = UfnD

6

4

2

0

ym

axD

(a)

9 12 15 180

10

20

30

40

50

60Voltage output

Ur = UfnD

V(V

)

R = 2e2Ω

R = 2e3ΩR = 2e4Ω

R = 2e5ΩR = 2e6Ω

(b)

00

04

08

12

Harvested power

R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

Ur = UfnD

P(W

)

9 12 15 18

(c)



References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











= 119866 (119877)119883 (18)

where

119883 = (

1198831

1198832

1198833

)

119866 (119877)119883

=(((

(

0 1 0

minus120596119899

2

minus(2120585120596119899minus1205881198802

1198631198861

2119898)

120579

119898

0 minus120579

119888119901

minus1

119877119888119901

)))

)

(19)



2=



112058221205823

= minus120596119899

2

119877119888119901 The first two






119892at 1205821=


119875 (119905) = (


119901119877)

1 + 12059621198772119888119901

2)

2

119877minus1

(20)



119903varied in





to 200






119903is equal



03

02

01

00

Non

dim

ensio

nal a

mpl

itude

s (AD

) U

L

D

3 6 9 12 15

Ur = UfnD


037 Blevins (1990)076 Blevins (1990)








119903= 64







119903= 75 when 119877 is




119899 The compu-


119903


119903= 5345)



119903=

578 and 119877 = 11198646Ω When 119877 = 11198645Ω and 119880119903= 596





006

003

000

minus003

minus006320 330 340 350 360 370

tT

yD

(a)

03

00

minus03

320 330 340 350 360 370

tT

CL

(b)03

00

minus03320 330 340 350 360 370

tT

Volta

ge (V

)

(c)

tT

yD

06

00

minus0625 26 27 28 29 30

(d)

tT

CL

06

00

minus0625 26 27 28 29 30

(e)

tT

Volta

ge (V

)

8

4

0

minus4

minus825 26 27 28 29 30

(f)

tT

yD

012

000

minus01230 40 50 60 70

(g)

tT

CL

04

00

minus0430 40 50 60 70

(h)

tT

Volta

ge (V

)

02

00

minus0230 40 50 60 70

(i)




119875har =1198812

rms119877

(21)


(a)

Y = 048D

(b)

(c)

Y = 048D

(d)


55 60 65 7000

01

02

03

04

05

AD


0

2

4

6

8

Ur = UfnD

Ymax = 048D

Vmax = 58V

Vrm

s(v

olts)






5 Conclusions



50

25

00

minus25

minus50650 655 660 665 670 675 680

AD

tT


00

01

02

03

Damping coefficient

Dam

ping

coeffi

cien

t

Natural frequency

42

43

44

45

(Hz)

103 104 105 106

R (ohms)

(a)

6

8

10

12

14

16


Ur=

Uf

nD

102 103 104 105 106 107

R (ohms)

(b)







Nomenclature



00

01

02

03

04y

maxD

R = 100Ω

54 56 58 60 62 64

Ur = UfnD

R = 1KΩ

R = 100 KΩ

R = 1MΩ

R = 10 KΩ

(a)

54 56 58 60 62 64

102

101

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

Ur = UfnD

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(b)

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

102 103 104 105 106

R (ohms)

Ur = 54

Ur = 56

Ur = 58

Ur = 60Ur = 62

(c)

5654 58 60 62 64

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

Ur = UfnD

P(w

att)

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(d)






Competing Interests


Acknowledgments




R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

9 12 15 18

Ur = UfnD

6

4

2

0

ym

axD

(a)

9 12 15 180

10

20

30

40

50

60Voltage output

Ur = UfnD

V(V

)

R = 2e2Ω

R = 2e3ΩR = 2e4Ω

R = 2e5ΩR = 2e6Ω

(b)

00

04

08

12

Harvested power

R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

Ur = UfnD

P(W

)

9 12 15 18

(c)



References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










03

02

01

00

Non

dim

ensio

nal a

mpl

itude

s (AD

) U

L

D

3 6 9 12 15

Ur = UfnD


037 Blevins (1990)076 Blevins (1990)








119903= 64







119903= 75 when 119877 is




119899 The compu-


119903


119903= 5345)



119903=

578 and 119877 = 11198646Ω When 119877 = 11198645Ω and 119880119903= 596





006

003

000

minus003

minus006320 330 340 350 360 370

tT

yD

(a)

03

00

minus03

320 330 340 350 360 370

tT

CL

(b)03

00

minus03320 330 340 350 360 370

tT

Volta

ge (V

)

(c)

tT

yD

06

00

minus0625 26 27 28 29 30

(d)

tT

CL

06

00

minus0625 26 27 28 29 30

(e)

tT

Volta

ge (V

)

8

4

0

minus4

minus825 26 27 28 29 30

(f)

tT

yD

012

000

minus01230 40 50 60 70

(g)

tT

CL

04

00

minus0430 40 50 60 70

(h)

tT

Volta

ge (V

)

02

00

minus0230 40 50 60 70

(i)




119875har =1198812

rms119877

(21)


(a)

Y = 048D

(b)

(c)

Y = 048D

(d)


55 60 65 7000

01

02

03

04

05

AD


0

2

4

6

8

Ur = UfnD

Ymax = 048D

Vmax = 58V

Vrm

s(v

olts)






5 Conclusions



50

25

00

minus25

minus50650 655 660 665 670 675 680

AD

tT


00

01

02

03

Damping coefficient

Dam

ping

coeffi

cien

t

Natural frequency

42

43

44

45

(Hz)

103 104 105 106

R (ohms)

(a)

6

8

10

12

14

16


Ur=

Uf

nD

102 103 104 105 106 107

R (ohms)

(b)







Nomenclature



00

01

02

03

04y

maxD

R = 100Ω

54 56 58 60 62 64

Ur = UfnD

R = 1KΩ

R = 100 KΩ

R = 1MΩ

R = 10 KΩ

(a)

54 56 58 60 62 64

102

101

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

Ur = UfnD

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(b)

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

102 103 104 105 106

R (ohms)

Ur = 54

Ur = 56

Ur = 58

Ur = 60Ur = 62

(c)

5654 58 60 62 64

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

Ur = UfnD

P(w

att)

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(d)






Competing Interests


Acknowledgments




R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

9 12 15 18

Ur = UfnD

6

4

2

0

ym

axD

(a)

9 12 15 180

10

20

30

40

50

60Voltage output

Ur = UfnD

V(V

)

R = 2e2Ω

R = 2e3ΩR = 2e4Ω

R = 2e5ΩR = 2e6Ω

(b)

00

04

08

12

Harvested power

R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

Ur = UfnD

P(W

)

9 12 15 18

(c)



References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










006

003

000

minus003

minus006320 330 340 350 360 370

tT

yD

(a)

03

00

minus03

320 330 340 350 360 370

tT

CL

(b)03

00

minus03320 330 340 350 360 370

tT

Volta

ge (V

)

(c)

tT

yD

06

00

minus0625 26 27 28 29 30

(d)

tT

CL

06

00

minus0625 26 27 28 29 30

(e)

tT

Volta

ge (V

)

8

4

0

minus4

minus825 26 27 28 29 30

(f)

tT

yD

012

000

minus01230 40 50 60 70

(g)

tT

CL

04

00

minus0430 40 50 60 70

(h)

tT

Volta

ge (V

)

02

00

minus0230 40 50 60 70

(i)




119875har =1198812

rms119877

(21)


(a)

Y = 048D

(b)

(c)

Y = 048D

(d)


55 60 65 7000

01

02

03

04

05

AD


0

2

4

6

8

Ur = UfnD

Ymax = 048D

Vmax = 58V

Vrm

s(v

olts)






5 Conclusions



50

25

00

minus25

minus50650 655 660 665 670 675 680

AD

tT


00

01

02

03

Damping coefficient

Dam

ping

coeffi

cien

t

Natural frequency

42

43

44

45

(Hz)

103 104 105 106

R (ohms)

(a)

6

8

10

12

14

16


Ur=

Uf

nD

102 103 104 105 106 107

R (ohms)

(b)







Nomenclature



00

01

02

03

04y

maxD

R = 100Ω

54 56 58 60 62 64

Ur = UfnD

R = 1KΩ

R = 100 KΩ

R = 1MΩ

R = 10 KΩ

(a)

54 56 58 60 62 64

102

101

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

Ur = UfnD

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(b)

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

102 103 104 105 106

R (ohms)

Ur = 54

Ur = 56

Ur = 58

Ur = 60Ur = 62

(c)

5654 58 60 62 64

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

Ur = UfnD

P(w

att)

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(d)






Competing Interests


Acknowledgments




R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

9 12 15 18

Ur = UfnD

6

4

2

0

ym

axD

(a)

9 12 15 180

10

20

30

40

50

60Voltage output

Ur = UfnD

V(V

)

R = 2e2Ω

R = 2e3ΩR = 2e4Ω

R = 2e5ΩR = 2e6Ω

(b)

00

04

08

12

Harvested power

R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

Ur = UfnD

P(W

)

9 12 15 18

(c)



References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a)

Y = 048D

(b)

(c)

Y = 048D

(d)


55 60 65 7000

01

02

03

04

05

AD


0

2

4

6

8

Ur = UfnD

Ymax = 048D

Vmax = 58V

Vrm

s(v

olts)






5 Conclusions



50

25

00

minus25

minus50650 655 660 665 670 675 680

AD

tT


00

01

02

03

Damping coefficient

Dam

ping

coeffi

cien

t

Natural frequency

42

43

44

45

(Hz)

103 104 105 106

R (ohms)

(a)

6

8

10

12

14

16


Ur=

Uf

nD

102 103 104 105 106 107

R (ohms)

(b)







Nomenclature



00

01

02

03

04y

maxD

R = 100Ω

54 56 58 60 62 64

Ur = UfnD

R = 1KΩ

R = 100 KΩ

R = 1MΩ

R = 10 KΩ

(a)

54 56 58 60 62 64

102

101

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

Ur = UfnD

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(b)

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

102 103 104 105 106

R (ohms)

Ur = 54

Ur = 56

Ur = 58

Ur = 60Ur = 62

(c)

5654 58 60 62 64

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

Ur = UfnD

P(w

att)

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(d)






Competing Interests


Acknowledgments




R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

9 12 15 18

Ur = UfnD

6

4

2

0

ym

axD

(a)

9 12 15 180

10

20

30

40

50

60Voltage output

Ur = UfnD

V(V

)

R = 2e2Ω

R = 2e3ΩR = 2e4Ω

R = 2e5ΩR = 2e6Ω

(b)

00

04

08

12

Harvested power

R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

Ur = UfnD

P(W

)

9 12 15 18

(c)



References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










50

25

00

minus25

minus50650 655 660 665 670 675 680

AD

tT


00

01

02

03

Damping coefficient

Dam

ping

coeffi

cien

t

Natural frequency

42

43

44

45

(Hz)

103 104 105 106

R (ohms)

(a)

6

8

10

12

14

16


Ur=

Uf

nD

102 103 104 105 106 107

R (ohms)

(b)







Nomenclature



00

01

02

03

04y

maxD

R = 100Ω

54 56 58 60 62 64

Ur = UfnD

R = 1KΩ

R = 100 KΩ

R = 1MΩ

R = 10 KΩ

(a)

54 56 58 60 62 64

102

101

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

Ur = UfnD

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(b)

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

102 103 104 105 106

R (ohms)

Ur = 54

Ur = 56

Ur = 58

Ur = 60Ur = 62

(c)

5654 58 60 62 64

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

Ur = UfnD

P(w

att)

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(d)






Competing Interests


Acknowledgments




R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

9 12 15 18

Ur = UfnD

6

4

2

0

ym

axD

(a)

9 12 15 180

10

20

30

40

50

60Voltage output

Ur = UfnD

V(V

)

R = 2e2Ω

R = 2e3ΩR = 2e4Ω

R = 2e5ΩR = 2e6Ω

(b)

00

04

08

12

Harvested power

R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

Ur = UfnD

P(W

)

9 12 15 18

(c)



References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00

01

02

03

04y

maxD

R = 100Ω

54 56 58 60 62 64

Ur = UfnD

R = 1KΩ

R = 100 KΩ

R = 1MΩ

R = 10 KΩ

(a)

54 56 58 60 62 64

102

101

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

Ur = UfnD

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(b)

100

10minus1

10minus2

10minus3

10minus4

Vrm

s(v

olts)

102 103 104 105 106

R (ohms)

Ur = 54

Ur = 56

Ur = 58

Ur = 60Ur = 62

(c)

5654 58 60 62 64

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

Ur = UfnD

P(w

att)

R = 100ΩR = 1KΩ

R = 100 KΩ

R = 1MΩ

(d)






Competing Interests


Acknowledgments




R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

9 12 15 18

Ur = UfnD

6

4

2

0

ym

axD

(a)

9 12 15 180

10

20

30

40

50

60Voltage output

Ur = UfnD

V(V

)

R = 2e2Ω

R = 2e3ΩR = 2e4Ω

R = 2e5ΩR = 2e6Ω

(b)

00

04

08

12

Harvested power

R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

Ur = UfnD

P(W

)

9 12 15 18

(c)



References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

9 12 15 18

Ur = UfnD

6

4

2

0

ym

axD

(a)

9 12 15 180

10

20

30

40

50

60Voltage output

Ur = UfnD

V(V

)

R = 2e2Ω

R = 2e3ΩR = 2e4Ω

R = 2e5ΩR = 2e6Ω

(b)

00

04

08

12

Harvested power

R = 2e2ΩR = 2e3Ω

R = 2e4Ω

R = 2e5ΩR = 2e6Ω

Ur = UfnD

P(W

)

9 12 15 18

(c)



References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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