Study on Fluid-Induced Vibration Power Harvesting of

Research ArticleStudy on Fluid-Induced Vibration Power Harvesting ofSquare Columns under Different Attack Angles

Meng Zhang1 Guifeng Zhao1 and Junlei Wang2

1School of Civil Engineering Zhengzhou University Zhengzhou 450001 China2School of Chemical Engineering and Energy Zhengzhou University Zhengzhou 450001 China

Correspondence should be addressed to Junlei Wang just4pipi126com

Received 4 April 2017 Revised 16 June 2017 Accepted 11 July 2017 Published 10 August 2017

Academic Editor Micol Todesco

Copyright copy 2017 Meng Zhang 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

A model of the flow-vibration-electrical circuit multiphysical coupling system for solving square column vortex-induced vibrationpiezoelectric energy harvesting (VIVPEH) is proposed in this paper The quasi steady state theory is adopted to describe the fluidsolid coupling process of vortex-induced vibration based on the finite volume method coupled Gauss equation The vibrationalresponse and the quasi steady state form of the output voltage are solved by means of the matrix coefficient method and interactivecomputing The results show that attack angles play an important role in the performance of square column VIVPEH of which120572 = 45∘ is a relatively ideal attack angle of square column VIVPEH

1 Introduction

Recently the development and utilization of new energysources have become a research hotspot among which thestudy of capturing energy from environment has receivedmuch more attention One of the most important ways ofenvironmental energy harvesting is capturing energy fromfluid which can be divided into two kinds wind energy andwater energy Most of the traditional wind power and hydro-electric power facilities use the rotating turbine device to har-vest energy with large volume device and low energy densityThemicro energy technology which can extract energy fromenvironment and convert it into electric energy [1ndash4] hasthe features of functional continuity small volume and highenergy density In the late 1990s the vibration piezoelectricenergy harvesting technology has beenwidely used to harvestenvironmental flow energy and convert it into vibrationenergy [5ndash9] which is a kind ofmicro energy technologywithcontinuous and nonconsuming energy supply Therefore itis an effective method for the microminiaturization of flowinduced vibration energy harvesting device

In fluid dynamics there is a potential physical phe-nomenon that can be used for energy harvesting called

vortex-induced vibration Vortex-induced vibration is thatwhen the fluid flows through the bluff body the formationand periodic shedding of the vortexwill cause the vibration ofthe bluff body Once the vibration intensity reaches a certainlevel the flow field shedding will be locked which results inlarge vibration energy In other words vortex-induced vibra-tion is a kind of periodic steady or unsteady fluid structureinteraction phenomenon which has the characteristics ofcontinuity and easy excitation [10 11]

It is a challenging work to solve the problem of vortex-induced vibration energy harvesting The key problem hereis how to transform the flow energy into vibration energyefficiently In recent years many meaningful research workshave been carried out on using vortex-induced vibration tocollect ocean energy and wind energy Among them theenergy conversion of circular bluff body piezoelectric vortex-induced vibration is mostly concerned Allen and Smits [12]have studied the theory of energy harvesting of piezoelectricmaterials and designed an ldquoeelrdquo energy harvesting modelwhich can be used to harvest the fluid kinetic energy inthe water tank On the basis of film theory the ldquoeelrdquo modeldevice can also be used to harvest the vortex shedding energyof ldquolock-inrdquo phenomenon Taylor et al [13] used the ldquoeelrdquo

HindawiGeofluidsVolume 2017 Article ID 6439401 18 pageshttpsdoiorg10115520176439401

2 Geofluids

device of a PVDF polymer to harvest marine energy whichwas placed in a water tank with the length of 2413mmthe width of 762mm and the thickness of 150 120583m Theresearch shows that when the flapping frequency of PVDFcloses to the vortex shedding frequency the energy collectionperformance will be improved the maximum voltage 3Vappears in the water flow velocity of 05ms In MarineRenewable Energy Laboratory (MRELab) at the Universityof Michigan Bernitsas et al [14 15] have presented a vortex-induced vibration for aquatic clean energy (VIVACE) toutilize the VIV phenomenon to generate power The latterstudies have been conducted in support of model tests forVIVACE converter which harnesses hydrokinetic energyenhancing flow induced motions (FIM) and particularlyVIV and various forms of galloping Lee and Bernitsas[16 17] built a devicesystem Vck to replace the physicaldampersprings of the VIVACE with virtual elements Thetesting was performed in the Low Turbulence Free SurfaceWater Channel of theUniversity ofMichigan at 40000 lt Re lt120000 and damping 0 lt 120585 lt 016 To continue and improvethe work of VIVACE which is used to convert hydrokineticenergy fromoceanriver currents to electricity Raghavan andBernitsas [18 19] conducted experiments in the regime rightbefore transition from laminar to turbulent flow Hysteresiswas not observed in the experiments because the parametersremain in the 2P domain Higher VIV amplitude of 21 to27 diameters was obtained and the range of synchronizationwas increased by increasing the damping in VIVACE Changet al [20] put a PTC device in the surface of the VIVACEconverter to enhance the viscous damping the results showthat proper PTC could enhance the conversion efficiency ofhydrokinetic energy to mechanical in VIVACE For solvingthe problem of VIV some excellent researches have alsobeen conducted in MRELab With the help of OpenFOAMsoftware Wu et al [21] have developed a CFD code to solvethe VIV problem of a single cylinder with PTC Ding etal [22] have also developed a CFD code in OpenFOAMto solve the VIV problem of multiple circular cylindersAbdelkefi et al [23 24] andMehmood et al [25] investigatedthe possibility of harvesting energy from elastically mountedcircular cylinders They used respectively a wake oscillatormodel and direct numerical simulations to determine thefluctuating lift force Abdelkefi et al [23] reported thatthe aerodynamic nonlinearity results in the presence ofhardening behavior Both research studies determined theeffects of the load resistance on the synchronization regionand level of the harvested power They demonstrated thatmaximum level of the harvested power can be associatedwith minimum values in the transverse displacement dueto the shunt damping effect To investigate the possibilityof harvesting energy from other flow induced vibrationsdifferent investigations have been performed Abdelkefi andNuhait [26] investigated the effects of cambered wing-basedpiezoaeroelastic energy harvesters on the flutter speed andthe performance of the harvester Dai et al [27] and Yan andAbdelkefi [28] investigated the potential of harvesting energyfrom a combination of base excitations and VIV or gallopingrespectively Robbins et al [29] proposed a device similarto the ldquoeelrdquo model consisting of piezoelectric film arrays

(single size 203 times 279 times 05mm) with different configurationsto collect wind energy generated by induced draft fan themaximum output power is up to 10mW Akaydin et al [30]proposed a structure (main size 30 times 16 times 02mm) whichcan separate the piezoelectric film from the bluff body inwhich the maximum output power is 4 120583W Gao et al [31]carried out an experiment model (unit piezoelectric elementsize is 58 times 10mm cylinder diameter is 291mm and cylinderlength is 36mm) in which the cylinder is connected withthe PZT cantilever beam and the PZT beam is fixed atone end in the wind tunnel Thus electrical charges wereaccompanied by the wind-induced vibration of the PZTcantilever beam The ldquolock-inrdquo phenomenon is observed inthe wind-induced vibration and a high voltage output isobtained in this region Akaydin et al [32] presented anenergy harvesting experiment in a wind tunnel in which thecylinder is connected with the free end of a cantilever beamwith aluminum shim device The results show that when thecircuit resistance is changed the natural frequency of thesystem will be changed also so that the maximum value ofthe electromechanical coupling damping can be obtainedThe above researches are mainly focused on the energycollection models of the cylindrical bluff body under vortex-induced vibration Studies on the vortex-induced vibrationandpiezoelectric energy harvesting of noncylindrical sectionsuch as square column and triangular column are relativelyless

It is very important to consider the different attack anglesin the analysis of square column vortex-induced vibrationbecause of the asymmetry The motion of the fluid at theedge of the square column is very irregular which willgreatly affect the vortex shedding near the column wall Thesquare column vortex shedding process is different fromthat of smooth circular cylinder there is a direct vortexshedding without a smooth transition region The boundarylayer separation point of the square column is fixed at thecorner of the leading edge without changing its position withthe varying of Reynolds number Therefore it is of greatpractical significance to study on the influence of differentattack angles of square column vortex-induced vibrationpiezoelectric energy harvesting (VIVPEH)

In this paper a model of square column VIVPEH ispresented which can be used to investigate the mechanismof different attack angles on the energy capturing efficiencyThe quasi steady state theory is adopted to describe the fluidsolid coupling process of vortex-induced vibration based onthe finite volume method coupled Gauss equation Finallythe performance of different attack angles on the VIVPEHis carried out with considering the flow field parameters thevibration characteristics and the energy collection parame-ters

2 Physical Model

The physical model of square column VIVPEH is shown inFigure 1 inwhich119880 stands for the inflowvelocity120572 stands forthe angle between the square column and the inflow velocity119872 is the mass of the column119870 is the elastic coefficient of the

Geofluids 3

R

U

PZT

(a) Side view of the system

R K C

Fy

FL

F

F

M

L0

U

D

x

(b) Mass-Spring-Damping-Resistanceload system

Figure 1 Physical model of square column energy harvesting system

system 119862 is the damping of the system and 119877 is the externalresistance load

3 Mathematical Model of Energy Harvesting

To analyze the coupling process of three different fieldsexternal flow field vortex-induced vibration and circuit thispaper uses the Navier-Stokes equation to describe the vortex-induced vibration uses a linear second-order differentialequation to describe the vortex-induced vibration of thesingle degree of freedom 119872-119862-119870 (mass spring damping)system and finally uses coupled Gauss law and vibrationequations to describe the electromechanical coupling sys-tem

31 Fluid Solid Coupling Model The external flow fieldis calculated by the continuity equation and the Navier-Stokes equation Flow simulations presented in this paperare produced by open source CFD tool OpenFOAM whichis composed of C++ libraries solving continuum mechanicsproblems with a finite volume discretization method Sup-pose the external flow field is 2D and unsteady The time-dependent viscous flow solutions can be obtained by numeri-cal approximation of the incompressible unsteady Reynolds-Averaged Navier-Stokes (URANS) equations in conjunctionwith the one-equation Spalart-Allmaras (S-A) turbulencemodel [33] where a second-order Gauss integration schemewith a linear interpolation is used in the governing equa-tions for the divergence gradient and Laplacian terms Fortime integration second-order backwards Euler method isemployed The numerical discretization scheme has second-order accuracy in space and time A pressure implicit withsplitting of operators (PISO) algorithm is used for solvingmomentumand continuity equations together in a segregatedway The equations of motion for the square column are

solved using a second-order mixed implicit and explicittime integration scheme The basic equations of URANSare

120597119880119894120597119909119894 = 0120597119880119894120597119905 + 119880119895

120597119880119894120597119909119895 = minus1120588120597119901120597119909119894 +

120597120597119909119895 (2]119878119894119895 minus 11990610158401198951199061015840119894) (1)

where 119901 is pressure 120588 is fluid density ] is dynamic viscosity119880119894 is the mean flow velocity vector and 119878119894119895 is the strain ratetensor

119878119894119895 = 12 (120597119880119894120597119909119895 +

120597119880119895120597119909119894 ) (2)

To solve the URANS equations for mean flow propertiesand potential turbulence flow the Boussinesq eddy-viscosityapproximation is adopted here which relates to the Reynoldsstress and the velocity gradient The quantity 12058811990610158401198951199061015840119894 is theReynolds stress tensor and can be modeled as 12058811990610158401198951199061015840119894 = 2120583119905119878119894119895where 120583119905 is the turbulence eddy viscosity

The S-A model is widely used for turbulence closureThe eddy-viscosity coefficient can be calculated from thefollowing transport equation

120597]120597119905 + 119906119895 120597]120597120594119895 = 1198881198871] minus 1198881199081119891119908 (]119889)2

+ 1120590 120597120597120594119895 [(] + ]) 120597]120597120594119895] + 1198881198872120597]120597120594119894

sdot 120597]120597120594119895

(3)

4 Geofluids

In S-A model the turbulence eddy viscosity 120583119905 can beobtained by

120583119905 = 120588]119891V1 (4)

in which 119891V1 = 1205943(1205943 + 1198883]1) and 120594 equiv ]] where 120594 is anintermediate value ] is working variable of the turbulencemodel and depends on the transport equation (3)

The details of the transport equation are in Spalart andAllmaras [33] and the trip terms 1198911199051 and 1198911199052 are switchedoff and a ldquotrip-lessrdquo initial condition was added for solvingworking variable ] This approach was successfully used inthe work of Wu et al [21] and Ding et al [22] for circularcylinders

The motion equations of a single degree of freedom119872-119862-119870 system can be represented by the following linearsecond-order differential equation

119872 + 119862 + 119870119884 = 119865119910 (5)

The relationships between119872 119870 and 119862 are as follows

120596119899 = radic 119870119872119862 = 2120585119872120596119899

(6)

where 119865119910 stands for the force on unit volume of the flow fieldwhich is perpendicular to the flow direction Y stands for thecolumn vibration displacement and represent the first-or second-order derivative of the column vibration displace-ment respectively 120596119899 is the natural circular frequency and 120585is the dimensionless damping ratioThe dynamic response ofthe column can be obtained by solving the motion equationsand the fluid governing equations simultaneously

32 Electromechanical Coupling Model In order to describethe relationship between the amplitude and the voltage in thevortex-induced vibration circuit Gauss law is adopted in thispaper Theoretical derivations are as follows

119872 + 119862 + 119870119884 minus 120579119881 = 119865119910 (7)

120579 + 119862119901 + 119881119877 = 0 (8)

where 120579 is the electromechanical coupling coefficient 119862119901 isthe capacitance coefficient and 119881 is the voltage

The influences of the vortex-induced vibration system onthe circuit output voltage have been considered in (7) and (8)respectively At the same time the negative feedback effect ofthe circuit on the vibration system is also taken into accountthat is to say the influence of electromechanical coupling isconsidered Combined with the flow field calculation resultswe can carry out the flow-mechanical-electrical couplinganalysis

In order to solve the damping and natural frequency ofthe system we use the matrix method to calculate the two-order nonhomogeneous ordinary differential equation (7)The homogeneous equation of (7) is as follows

119872 + 119862 + 119870119884 minus 120579119881 = 0 (9)

Let 1198831 = 119884 1198832 = and 1198833 = 119881 substitute (6) into (8) and(9) we can get

1 = 11988322 = minus12059611989921198831 minus 21205851205961198991198832 + 12057911987211988333 = minus 1205791198621199011198832 minus

1198833119877119862119901 (10)

The above equations can be expressed in the followingmatrixform

= 119861 (119877)119883 (11)

where

119883 = [1198831 1198832 1198833]119879

119861 (119877) =[[[[[[[

0 1 0minus1205961198992 minus2120585120596119899 1205791198720 minus 120579119862119901 minus 1119877119862119901

]]]]]]] (12)

The matrix 119861(119877) has three different eigenvalues of 119896119894 ofwhich 119894 = 1 2 3 Spalart and Allmaras [33] have pointedout that the first two eigenvalues are similar to the ones ofvibration system without circuit and yet the third eigenvalueis associated with the electromechanical coupling effect suchas piezoelectric system affected by the foundation or theaeroelastic excitation and is negative constant There areconjugate relations between 1198961 and 1198962 in which the real partand the imaginary part of the conjugate solution stand forthe damping and natural frequency of the electromechanicalcoupling system respectively Given that 1198963 is negativeconstant we consider only the real part of 1198961 and 1198962 whencomputing the trivial solution of the matrix 119861(119877)4 Quasi Steady State Model forOutput Voltage

In this section we adopt the quasi steady state modelproposed by Barrero-Gil et al [34] to describe the amplitudeof vortex-induced vibration and calculate the time-varyingvibration energy harvestingWhen the vortex-induced vibra-tion is in the synchronization region the vibration amplitudeof the system can be expressed as the following sine function

119884 = 119884max sin (120596119899119905) (13)

where 119884max is the maximum column vibration displacementIt is worth noting that the voltage time-history curve and

the vibration amplitude time-history curve are synchronousthat is to say there is no phase difference Substituting (13)into (11) we can get the analytical solution of the quasi steadystate voltage with MATLAB

119881 (119905) = 120579120596119899119877119884max1 + 120596119899211987721198621199012 (119890minus119905119877119862119901 minus cos (120596119899119905)

minus 120596119899119877119862119901 sin (120596119899119905)) (14)

Geofluids 5

Table 1 Calculation parameters for square column VIVPEH system

Symbols and units Physical meanings Numerical values119872 [kg] Square column mass 02979119870 [Nm] Elastic coefficient of the system 579sim584119862 [Nsdotsm] Damping of the system 00325sim045119863squ [m] Square column side length 00016120572 [degree] Angle between the square column and the inflow velocity 0sim75120577 Damping ratio of the system 000121119891119899 [Hz] Natural frequency of the system 7012sim7132 (water)119862119901 [nF] Capacitance 120120579 Electromechanical coupling coefficient 155 times 10minus3

120583 [Pasdots] Dynamic viscosity 00011379] [m2s] Kinematic viscosity 1139 times 10minus6 (water)120588 [kgm3] Density 9991026 (water)

The corresponding output power can be obtained by thefollowing equation

119875 (119905) = 1198812 (119905)119877 (15)

The above mathematical expressions include the solvingprocess of fluid solid coupling and electromechanical cou-pling

First of all we can get the flow field pressures 119875 and 119865119910by solving the URANS equations in OpenFOAM Secondlywe can obtain the column vibration displacement 119884 thedamping 119862 and natural circular frequency 120596119899 by solving (5)to (11) It should be noted that the motion of the columnis influenced by the pressure of the flow field At the sametime the vibration of the column gives feedback to the flowfield and causes the change of flow field distribution Thusthe fluid solid coupling problem can be solved by interactivecomputing Finally the time-history curve of output voltageand output power can be obtained by (14) and (15) Theabove is the whole computing process of flow-vibration-electromechanical coupling system

5 Numerical Results

51 Analysis Parameters and Cases To carry out the numer-ical calculation of vortex-induced vibration this paper pro-vides the parameters of the vibration energy harvestingsystem as shown in Table 1 in which119863squ is the side length ofthe square column 119863Nor stands for the dimensionless char-acteristic length of the square column and the relationshipbetween119863Nor and119863squ is given in

119863Nor = 119863squ (sin120572 + cos120572) (16)

Based on the parameters of Table 1 and formula (16) we cancalculate the numerical values of 119863Nor as 00016 under 0∘attack angle 000196 under 15∘ attack angle 000219 under 30∘attack angle 000226 under 45∘ attack angle 000219 under60∘ attack angle and 000196 under 75∘ attack angle

The computational grid and boundary conditions ofsquare column vortex-induced vibration under different

attack angles are given in Figure 2 where the computationaldomain is 20 times 20D and the entire domain includes fiveboundaries velocity inlet velocity outlet top bottom anda column wall The inlet velocity is considered as uniformand constant velocity For outlet boundary a zero gradientcondition is specified for velocity The top and bottom con-dition are defined as a wall boundary In present numericalstudy a moving wall boundary condition is applied forthe square column when the column is in VIV The two-dimensional structured grids were generated with the helpof ldquoGambitrdquo software The grid domain size is 20 times 20D Thesquare column was set in the center in the domain to ensurethat the results of the numerical model are accurate Theconditions at the outlet are close to the assumed conditionsThe computational domain in the vicinity of each cylinder is a3 times 3D square where the grid density for the near-wall regionis enhanced to solve for high resolution in flow properties

When the external resistance value is 119877 = 1 times 106Ω thespring stiffness value of the square column VIVPEH systemis 119870squ = 580Nm and the corresponding damping value is119862squ = 02Nsm Taking the range values of the flow velocityas 003927ms to 008975ms we can calculate the numericalvalues of 119880119903squ under different attack angles as shown inTable 2 in which 119880119903squ is the reduction velocity defined asfollows

119880119903squ = 119880119891119899119863squ (17)

where 119891119899 = 1205961198992120587In this paper the matrix method is used to evaluate the

influence of the external resistance load on the damping andnatural frequency of the electromechanical coupling systemby MATLAB Then the damping and natural frequencyvalues can be used for initial conditions in OpenFOAMto compute the vibration amplitude of the vortex-inducedvibration system under different Reynolds numbers (94 lt119877119890 lt 115) The main focus of this paper is to investigate theinfluence of external resistance load (119877 = 1 times 103Ω 1 times 104Ω1 times 105Ω 1 times 106Ω 1 times 107Ω) on the energy conversion ofthe square column VIVPEH system including the effect onvibration amplitude output voltage and power

6 Geofluids

(a) 120572 = 0∘ (b) 120572 = 15∘ (c) 120572 = 30∘

(d) 120572 = 45∘ (e) 120572 = 60∘ (f) 120572 = 75∘

Square column

Top

y

x

OutletInlet

Bottom

(g)

Figure 2 Computational grid and boundary conditions

52 System Damping and Natural Frequency CharacteristicsThe damping and natural frequency of the electromechanicalcoupling system can be obtained by MATLAB and the realand imaginary parts of the circuit conjugate solutions areshown in Figure 3

According to Figure 3 we can see that the total dampingof the system is small when the resistance load is smallIn the case of 119877 lt 1 times 105Ω the system total dampingis increased with the resistance load increasing Once the

resistance load 119877 reaches 1 times 105Ω the system total dampingreaches the maximum value It is noteworthy that the totaldamping of the system decreases instead of increasing whenthe resistance load keeps increasing that is 119877 gt 1 times 105ΩFor the natural circular frequency when 119877 lt 3 times 104Ω thefrequency value remains at 44 rads when 119877 gt 2 times 106Ω thefrequency value approximately remains at 50 rads Generallythe natural circular frequency value of the system is relativelystable which is kept in the range of 44ndash50 rads

Geofluids 7

Table 2 Run case of square column in OpenFOAM

119880 (ms) 119880119903squ120572 = 0∘ 120572 = 15∘ (120572 = 75∘) 120572 = 30∘ (120572 = 60∘) 120572 = 45∘003927 35 285784 256223 247525004488 4 326611 292826 282885005049 45 367437 329429 31824600561 5 408263 366032 353607005834 52 424594 380673 367751006059 54 440924 395315 381895006283 56 457255 409956 39604006732 6 489916 439239 424328007293 65 530742 475842 459689007854 7 571569 512445 49505008415 75 612395 549048 53041008976 8 653221 585652 565771009537 85 694048 622255 601132

Real Imaginary

00

05

10

15

20

25

30

35

Real

103 104 105 106 107102

R (ohms)

44

45

46

47

48

49

50

Imag

inar

y

Figure 3 Real and imaginary parts of the circuit conjugate solu-tions

53 Vibration Characteristics of Square Column VIVPEHunder Different Attack Angles In the following numericalsimulations we take the resistance load 119877 = 1 times 106Ωand compute the amplitude response of the square columnVIVPEH under different attack angles and different flowvelocities as shown in Figure 4 Note here that the naturalfrequency of square column is 119891119899 = 698Hz when theresistance load is 119877 = 1 times 106Ω The dimensionless vibrationamplitude 119884squ119863squ is adopted to indicate the vibrationresponse of the square column The maximum value of 119884squcan be obtained by means of averaging the peak value of atleast 60 displacement time-history response curves In thispaper the vibration amplitude of the smooth circular columnis provided to compare with that of the square column

As we can see in Figure 4 different attack angles obvi-ously have an effect on the peak vibration amplitude andlock-in region Moreover we can observe the ldquopresynchro-nizationrdquo ldquosynchronizationrdquo and ldquopostsynchronizationrdquo ofvortex-induced vibration curves of square column VIVPEHunder different attack angles

00

01

02

03

04

5 83 74 6UrSqu

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

DSq

uY

Squ

Figure 4 Dimensionless vibration amplitude of square columnVIVPEH under different attack angles and different flow velocities

531 120572 = 0∘ The lock-in region is from 119880119903squ = 63 to theend of 119880119903squ = 67 and the maximum amplitude is 119884squmax119863squ = 005 which appeared at 119880119903squ = 65 as shown inFigure 4 In order to see more clearly about the vibrationamplitude 119884squ119863squ some displacement time-history curvesand FFT analysis results for 120572 = 0∘are given in Figure 5It can be seen that when 119880119903squ is small the amplitude oftime-history curve presents a stable sinusoidal curve and themaximum amplitude is so small that it can be ignored whichmeans that there is almost no vortex-induced vibration in thesystemWith the increases of119880119903squ the systementers ldquopresyn-chronizationrdquo phase and the vibration amplitude 119884squ119863squgradually increases When the vortex shedding frequencyapproaches the natural frequency of square columnVIVPEHthe oscillation amplitude of the square column VIVPEH

8 Geofluids

minus001

000

001

6 8 10 124Time (s)

DSq

uY

Squ

(a) Nondimensional displacement for119880119903squ = 60

0

200

400

600

800

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)

(b) FFT analysis for119880119903squ = 60

minus006

minus004

minus002

000DSq

uY

Squ

002

004

006

18 21 24 27 3015Time (s)

(c) Nondimensional displacement for119880119903squ = 65

0

1000

2000

3000M

agni

tude

4 5 6 7 8 9 103Frequency (Hz)

(d) FFT analysis for119880119903squ = 65

Figure 5 Displacement time-history and FFT analysis results for 120572 = 0∘

increases significantly and the system enters synchronousphase which means the phenomenon of ldquolock-inrdquo occursUnder synchronous phase the vortex shedding frequency iskept constant and the amplitude of the system will remain ata higher value when the flow velocity 119880119903squ is increased from119880119903squ = 63 to the end of 119880119903squ = 67532 120572 = 15∘ It can be seen from Figure 2(b) that thereis a corner at the upper windward of the square columnwhich can obviously affect the flow field As is shown inFigure 5 the maximum amplitude decreases and the lock-in region is narrow (from 119880119903squ = 54 to 119880119903squ = 57) Themaximum amplitude is119884squmax119863squ = 0148 which appearedat119880119903squ = 57The displacement time-history curves and FFTanalysis results for 120572 = 15∘ are given in Figure 6 Similar withthe results of 120572 = 0∘ the displacement time-history curvecan also be divided into the following four stages ldquounsyn-chronizationrdquo ldquopresynchronizationrdquo ldquosynchronizationrdquo andldquopostsynchronizationrdquo

533 120572 = 30∘ The results in Figure 4 show that the max-imum amplitude of the square column is obviously increased

with a wider range of lock-in region (from 119880119903squ = 45to 119880119903squ = 57) the maximum amplitude 119884squmax119863squreaches 041 Similarly some of the displacement time-historycurves and FFT analysis results for 120572 = 30∘ are given inFigure 7The results in Figure 7(a) show that the whole time-history curve appears as a parabolic shape and the growthrate decreased gradually to a stable level at late stage whichindicates the coupling of the vortex shedding frequency andnatural frequency that is the lock-in phenomenon occursIt can also be seen that there is a ldquobeatrdquo phenomenon inthe amplitude curves as shown in Figure 7(a) There arethree peaks in the frequency spectrum curve as shown inFigure 7(b) which indicates that three harmonics occur inthe system The explaining of the above phenomena is asfollows In the process of vortex-induced vibration of thesquare column the vortex shedding frequency couples withthe natural frequency when the flow velocity 119880119903squ reaches acertain valueThe flow field near the square column surface isstrongly disturbed because of the corner point of the squarecolumn which results in the superposition of three vibrationfrequencies This is because the unbalance of the upper andlower aerodynamic force is caused by the obvious asymmetry

Geofluids 9

minus003

minus002

minus001

000DSq

uY

Squ

001

002

003

18 21 24 2715Time (s)


0

200

400

600

800

1000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000DSq

uY

Squ

005

010

015

18 21 2415Time (s)


0

2000

4000

6000

8000

10000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)



under attack angle 120572 = 30∘ When the flow velocity is about119880119903squ = 549 as shown in Figures 7(c) and 7(d) the amplitudecurve appears as a complete sine curve without noise Themaximum amplitude 119884squmax119863squ is increased to about 041and the vibration frequency is stabilized at 698Hz which canbe regarded as the best working condition of square columnVIVPEH

534 120572 = 45∘ As is shown in Figure 4 the displacementtime-history curve can also be divided into the follow-ing four stages ldquounsynchronizationrdquo ldquopresynchronizationrdquoldquosynchronizationrdquo and ldquopostsynchronizationrdquo The maxi-mum amplitude is 119884squmax119863squ = 028 and the lock-inregion is from 119880119903squ = 42 to the end of 119880119903squ = 54which appears earlier than that of other attack angles Thedisplacement time-history curves and FFT analysis resultsfor 120572 = 45∘ are given in Figure 8 It can be seen that thereis no severe aerodynamic disturbance which shows that thevortex-induced vibration response of symmetric bluff body isrelatively stable

535 120572 = 60∘ The displacement time-history curves andFFT analysis results for 120572 = 60∘ are given in Figure 9 which

are similar in shape to the one for 120572 = 30∘ with only slightdifferences in values According to the displacement time-history curves as shown in Figures 7 and 9 it can be seen thatthe amplitude results are almost exactly the same as the onefor 120572 = 30∘ in the lock-in region The same conclusion canalso be obtained from the spectral analysis results that is thephenomena of harmonic and noise are similarThis is becausethe spring force of the system and the gravity of the columnprovide a balance in the flowfield So it can be considered thatthe above two cases of 120572 = 30∘ and 120572 = 60∘ are symmetric

536120572 = 75∘ Theresults in Figure 10 show the displacementtime-history curves and FFT analysis results for 120572 = 75∘ Itis easy to see that both the displacement time-history curvesand the spectral analysis results are similar to those of 120572 = 15∘case which indicates that the cases of 120572 = 15∘ and 120572 = 75∘are also symmetric

In order to verify the above conclusions the StrouhalNumbers for the following four cases 120572 = 15∘ 120572 = 30∘120572 = 60∘ and 120572 = 75∘ are compared with each other as shownin Table 3 It can be seen that the Strouhal Number results forthe case of 120572 = 15∘ (120572 = 30∘) are almost equal to those for

10 Geofluids

minus02

minus01

00DSq

uY

Squ

01

02

3 6 9 12 15 18 21 24 27 300Time (s)


0

2000

4000

6000

8000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus03

minus02

minus01

00

01

02

03

04

DSq

uY

Squ

3 6 9 12 15 18 21 24 27 300Time (s)


0

10000

20000

30000

40000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)



Table 3 Comparison of Strouhal Number with different attackangle

U (ms) Strouhal Number15∘ 75∘ 30∘ 60∘

005049 01682 01682 01832 0183100561 01714 01714 01873 01872006278 01743 01742 01896 01896007293 01746 01746 01922 01922

the case of 120572 = 75∘ (120572 = 60∘) which indicates that the aboveanalysis is correct

To show the difference more clearly of the lock-in regionof square column VIVPEH under different attack angles wechoose the dimensionless frequency 119891squ119891119899squ to indicatethe frequency characteristics of square column VIVPEHwhere 119891squ is the vortex shedding frequency of the systemwhich can be obtained by Fast Fourier Transform (FFT)of the displacement time-history curve shown in Figures5ndash10 119891119899squ is the natural frequency of the square columnThe results in Figure 11 show the different lock-in regionsof square column VIVPEH under different attack angles

For the case of 120572 = 0∘ the range of values for the lock-inregion is 65 to 70 (65 le 119880119903squ le 70) the correspondingbandwidth value is 05 For the case of 120572 = 15∘ and 120572 =75∘ the range of values for the lock-in region is 54 to 57(54 le 119880119903squ le 57) the corresponding bandwidth valueis just 03 For the case of 120572 = 30∘ and 120572 = 60∘ therange of values for the lock-in region is 46 to 55 (46 le119880119903squ le 55) the corresponding bandwidth value is 09which is significantly larger than that of the above two casesWhen the attack angle is 120572 = 45∘ the range of values forthe lock-in region is 42 to 54 (42 le 119880119903squ le 54) thecorresponding bandwidth value increases to 12 In additionthe three different branch types of square column VIVPEHcan be observed in Figure 11 such as ldquopresynchronizationrdquoldquosynchronizationrdquo and ldquopostsynchronizationrdquo

Based on the above analysis results we believe that thecalculation parameters and response parameters for the caseof 120572 = 15∘ and 120572 = 30∘ are about the same as those for thecase of 120572 = 75∘ and 120572 = 60∘54 Phase Angle Analysis of Square Column VIVPEH underDifferent Attack Angles The analysis results of the initialphase angle and the synchronous phase angle of square

Geofluids 11

minus0006

minus0004

minus0002

0000

0002

0004

0006D

Squ

YSq

u

1815 21 2724 3330Time (s)


0

100

200

300

400

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus03

minus02

minus01

00

01

02

03

DSq

uY

Squ

18 20 2216Time (s)


0

5000

10000

15000

20000

25000

30000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)



column VIVPEH under different attack angles are given inFigures 12ndash15 It can be seen that the vibration is stable inthe initial stage with a single amplitude and frequency whichis completely determined by the vortex shedding frequencyTherefore the displacement time-history curve is almostsynchronous with the lift coefficient curve without phasedelay This is because the amplitude is small so that there isalmost no stagnation when the vibration amplitude reachesits maximum or minimum value Different from the initialvibration state the vortex shedding frequency is locked againin the synchronous state resulting in a multiple relationshipbetween the vibration frequency and the natural frequencyThat means in synchronization region there exists no phasedifference between the vortex shedding frequency and vibra-tion frequency Accordingly the displacement time-historycurve is completely synchronized with the lift coefficientcurve with some phase delay

Figure 16 shows that the vortex-induced vibration ofsquare column VIVPEH will stagnate for some time at thewave crest or the wave trough because of the buffering effectof spring As is shown in Figure 16(a) the vortices of 1198811 and1198812 move forward to a distance of 1198781 when the wave crests ofthe two adjacent steps appear At the moment 1198811 obviously

becomes thinner and longer while the vibration of the squarecolumn is still in the wave crest Similarly the vortices of1198813 and 1198814 move forward to a distance of 1198782 when the wavetroughs of the two adjacent steps appear It should be pointedout that the vibration of the square column is always in thewave crest or the wave trough which leads to the generationof the phase angle

55 Analysis of Near Wake Vortex Shedding of Square ColumnVIVPEH under Different Attack Angles The shapes of wakevortices in synchronized state of square column VIVPEHunder different attack angles are given in Figures 17ndash20 inwhich 119879 is the vibration period with subscript representingthe value of an attack angle The direction of the negativevorticity region is counterclockwise which is expressed inblue while the direction of the positive one is clockwisewhich is expressed in red It can be seen that for the case of120572 = 0∘ the flow field around the square column is stable thevibration amplitude is small and the wake vortex structurepresents a regular 2S shape In other words a positive andnegative vortex pair sheds in a cycle For 120572 = 15∘120572 = 75∘the position of near wake vortices shedding of square columnVIVPEHbeganmoving forward to the near columnwall with

12 Geofluids

minus02

minus01

00

01

02

DSq

uY

Squ

5 10 15 20 25 30 35 400Time (s)


0

5000

10000

15000

20000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus02

00

02

04

DSq

uY

Squ

3 6 9 12 150Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

2000

4000

6000

8000

10000

12000

14000

16000

Mag

nitu

de



increasing vibration amplitudes Correspondingly the wakevortex shedding array gradually changes its shape from 2Sshape to a circle As for 120572 = 30∘120572 = 60∘ it can be observedthat the wake vortex shedding mode changes obviously withan increasing width of the vortex array due to the increaseof vibration amplitude and the shape of the vortex pair ischanged from a flat shape to a regular elliptical shape Whenthe attack angle is 120572 = 45∘ the vibration amplitude decreaseswhich results in the wake vortex shedding mode changingback to the stable 2S mode

56 Voltage Output and Power Output of Square ColumnVIVPEH under Different Attack Angles Due to the influenceof different attack angles the vortex separation point ofsquare column VIVPEH is different from that of cylinderwhich results in the following analysis being more compli-cated Considering the symmetric nature of the case 120572 =15∘120572 = 75∘ and the case 120572 = 30∘120572 = 60∘ we take only thecase of 120572 = 0∘ 120572 = 15∘ 120572 = 30∘ and 120572 = 45∘ in the followinganalysis of voltage output When the resistance load is 119877 = 1times 106Ω the output voltage of the system can be calculated by(14)

The purpose of this section is to investigate themaximumvalue of the output voltage and the lock-in region to select theoptimal attack angle of square column VIVPEH The resultsof the maximum output voltage of square column VIVPEHunder different attack angles and the effective working area ofsynchronization are given in Figure 21 It can be seen that themaximum output voltage of square columnVIVPEH appearsat the case of 120572 = 45∘ and the corresponding value is 6732Vwhile the minimum value appears at the case of 120572 = 15∘120572 =75∘ which indicates that these two cases are not suitable forenergy harvesting Similarly the output voltage of the case of120572 = 0∘ is slightly higher than that of the case of 120572 = 15∘120572 =75∘ however its value is still small When the attack angleincreases from 120572 = 15∘ to 120572 = 45∘ the output voltage ofsquare column VIVPEH system increases to its maximumvalueThe results of the working region of synchronization inFigure 21 show that the total bandwidth of the above six casesis (42ndash57 53ndash57) and there are about 06 bandwidth of theregion of nonsynchronization The maximum bandwidth is(42ndash54) under attack angle 120572 = 45∘ which is higher thanthat of all other attack angle cases Accordingly the outputpower of the system can be calculated by (15)

Geofluids 13

minus02

minus01

00

01

02

DSq

uY

Squ

18 21 24 27 30 3315Time (s)


0

500

1000

1500

2000

2500

3000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000

005

010

015

DSq

uY

Squ

18 21 24 27 3015Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

4700

9400

14100

18800M

agni

tude



06

08

10

12

85 6 743

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

f３ＫＯf

N３ＫＯ

f３ＫＯfN３ＫＯ = 1

UrSqu

Figure 11 Nondimensional frequency of square column VIVPEH under different attack angles

14 Geofluids

minus0004

minus0003

minus0002

minus0001

0000

0001

0002

0003

0004

YD

minus15

minus10

minus05

00

05

10

15

CL

158154 156 160150 152

Time (s)

(a) Initial phase angle

minus006

minus004

minus002

000

002

004

006

YD

158154 156 160150 152

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

(b) Synchronous phase angle

Figure 12 Results of phase angle in initial and synchronization state for 120572 = 0∘

minus0004

minus0002

0000

0002

0004

YD

152 154 160158156150

Time (s)

minus10

minus05

00

05

10

CL


minus015

minus010

minus005

000

005

010

015YD

152 154 156 158 160150

Time (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

CL


Figure 13 Results of phase angle in initial and synchronization state for 120572 = 15∘120572 = 75∘

minus0002

0000

0002

YD

151 158157156 159150 154153152 160155

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus0004

0004


minus04

minus03

minus02

minus01

00

01

02

03

04

YD

151 152 159154 158156 157 160150 153 155

Time (s)

minus25

minus20

minus15

minus10

minus05

00

05

10

15C

L



Geofluids 15

minus0006

minus0004

minus0002

0000

0002

0004

0006YD

151 152 153 154 155 156 157 158 159 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL


151 155153 157 159156152 158154 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus02

minus01

00

01

02

YD



V1

V1

V2

V2

S1

(a) 119884max+ (wave crest)

V3

V4

S2

V3

V4

(b) 119884maxminus (wave trough)

Figure 16 Diagram of phase angle between displacement and lift force development history

tT３ＫＯ0 = 0 tT３ＫＯ0 = 0249

tT３ＫＯ0 = 0502 tT３ＫＯ0 = 075

Figure 17 Vortex structures in synchronization region for 120572 = 0∘

The results of the maximum output power of squarecolumn VIVPEH under different attack angles are givenin Figure 22 It can also be seen that the maximum valueof the output power appears at the case of 120572 = 45∘ and

tT３ＫＯ15 = 0 tT３ＫＯ15 = 025

tT３ＫＯ15 = 0497 tT３ＫＯ15 = 0752

Figure 18 Vortex structures in synchronization region for 120572 =15∘120572 = 75∘

the corresponding value is 45 times 10minus5W Therefore squarecolumn VIVPEH under attack angle 120572 = 45∘ is an ideal PEHbecause of its higher voltage output value and larger workingbandwidth

16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions

The energy harvesting features of square column VIVPEHunder different attack angles are investigated in this paperwith considering the vibration characteristics phase charac-teristics the near wake vortex sheddingmode and the outputvoltage and power of the systemThemain conclusions are asfollows

(1) Within the range of reduced velocity studied in thispaper the vortex-induced vibration curves of squarecolumn VIVPEH under different attack angles can beobtained which contains the phenomena of ldquopresyn-chronizationrdquo ldquosynchronizationrdquo and ldquopostsynchro-nizationrdquo The vortex shedding shape of square col-umn VIVPEH is dominated by 2S mode

(2) The attack angle has significant effect on the maxi-mum value of vibration amplitude of square columnand the lock-in region Due to the influence ofdifferent attack angles the boundary layer separationpoint does not move backwards like the cylinderThe maximum vibration amplitude and the lock-invibration region of square column will also fluctuateWhen the attack angle is equal to 45 degrees thesynchronization region can reach a value of 12 timesof reduction velocity

(3) The numerical analysis shows that the cases of 120572 =15∘120572 = 30∘ and 120572 = 75∘120572 = 60∘ for squarecolumnVIVPEH are symmetric which indicates that

(63ndash67)(54ndash57)

(andashb) is the Ur rangeof the synchronization

effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)

Figure 21 Comparison of maximum value of voltage output andsynchronization effective area between VIVPEH with differentshapes

00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (

Figure 22 Comparison of maximum value of power outputbetween VIVPEH with different shapes

the calculated results of the case of 120572 = 15∘120572 = 30∘are equal to the results of case of 120572 = 75∘120572 = 60∘respectively

(4) The maximum value of the output power is similarto that of the output voltage which appears at thecase of 120572 = 45∘ and the corresponding value is 45times 10minus5W and 6732V respectively Therefore 120572 =45∘ is a relatively ideal attack angle of square columnVIVPEH

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research has been funded by the National NaturalScience Foundation of China (Grants nos 51606171 51578512and 51108425) and the Outstanding Young Talent ResearchFund of Zhengzhou University (Grant no 1521322004)

Geofluids 17

References

[1] K A Cook-Chennault NThambi and AM Sastry ldquoPoweringMEMS portable devicesmdasha review of non-regenerative andregenerative power supply systems with special emphasis onpiezoelectric energy harvesting systemsrdquo Smart Materials andStructures vol 17 no 4 pp 1240ndash1246 2008

[2] J A Paradiso and T Starner ldquoEnergy scavenging formobile andwireless electronicsrdquo IEEE Pervasive Computing vol 4 no 1 pp18ndash27 2005

[3] N S Shenck and J A Paradiso ldquoEnergy scavenging with shoe-mounted piezoelectricsrdquo IEEE Micro vol 21 no 3 pp 30ndash422001

[4] G K Ottman H F Hofmann A C Bhatt and G A LesieutreldquoAdaptive piezoelectric energy harvesting circuit for wirelessremote power supplyrdquo IEEE Transactions on Power Electronicsvol 17 no 5 pp 669ndash676 2002

[5] S P Beeby M J Tudor and N M White ldquoEnergy harvestingvibration sources for microsystems applicationsrdquoMeasurementScience and Technology vol 17 no 12 pp R175ndashR195 2006

[6] S Roundy P K Wright and J Rabaey ldquoA study of lowlevel vibrations as a power source for wireless sensor nodesrdquoComputer Communications vol 26 no 11 pp 1131ndash1144 2003

[7] J Wang J Ran and Z Zhang ldquoEnergy harvester basedon the synchronization phenomenon of a circular cylinderrdquoMathematical Problems in Engineering vol 2014 Article ID567357 9 pages 2014

[8] J Wang S Wen X Zhao M Zhang and J Ran ldquoPiezoelectricWind Energy Harvesting from Self-Excited Vibration of SquareCylinderrdquo Journal of Sensors vol 2016 Article ID 2353517 12pages 2016

[9] M Zhang and J Wang ldquoExperimental study on piezoelectricenergy harvesting from vortex-induced vibrations and wake-induced vibrationsrdquo Journal of Sensors vol 2016 Article ID2673292 7 pages 2016

[10] C H K Williamson ldquoVortex dynamics in the cylinder wakerdquoAnnual Review of Fluid Mechanics vol 28 pp 477ndash539 1996

[11] C H Williamson and R Govardhan ldquoVortex-induced vibra-tionsrdquo Annual Review of Fluid Mechanics vol 36 no 1 pp 413ndash455 2004

[12] J J Allen and A J Smits ldquoEnergy harvesting eelrdquo Journal ofFluids and Structures vol 15 no s3-s4 pp 629ndash640 2001

[13] G W Taylor J R Burns S M Kammann W B Powers andT R Welsh ldquoThe energy harvesting Eel a small subsurfaceoceanriver power generatorrdquo IEEE Journal of Oceanic Engineer-ing vol 26 no 4 pp 539ndash547 2001

[14] M M Bernitsas K Raghavan Y Ben-Simon and E MH Garcia ldquoVIVACE (vortex induced vibration aquatic cleanenergy) a new concept in generation of clean and renewableenergy fromfluid flowrdquo Journal of OffshoreMechanics andArcticEngineering vol 130 no 4 Article ID 041101 15 pages 2008

[15] M M Bernitsas Y Ben-Simon K Raghavan and E M HGarcia ldquoThe VIVACE converter model tests at high dampingand reynolds number around 105rdquo Journal of OffshoreMechanicsand Arctic Engineering vol 131 no 1 pp 1ndash12 2009

[16] J H Lee N Xiros and M M Bernitsas ldquoVirtual damper-spring system for VIV experiments and hydrokinetic energyconversionrdquoOceanEngineering vol 38 no 16 pp 732ndash747 2011

[17] 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

[18] 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

[19] K Raghavan and M M Bernitsas ldquoEnhancement of highdamping VIV through roughness distribution for energy har-nessing at 8 times 103 lt Re lt 15 times 105rdquo in Proceedings of the27th International Conference on Offshore Mechanics and ArcticEngineering OMAE rsquo08 pp 871ndash882 June 2008

[20] C-C Chang R Ajith Kumar and M M Bernitsas ldquoVIV andgalloping of single circular cylinder with surface roughness at30 times 104 le Re le 12 times 105rdquoOcean Engineering vol 38 no 16 pp1713ndash1732 2011

[21] W Wu M M Bernitsas and K Maki ldquoSimulation vs experi-ments of flow induced motion of circular cylinder with passiveturbulence control at 35000 le Re le 130000rdquo in Proceedings ofthe ASME 2011 30th International Conference onOcean Offshoreand Arctic Engineering vol 136 pp 548ndash557 American Societyof Mechanical Engineers Rotterdam Netherlands June 2011

[22] L Ding M M Bernitsas and E S Kim ldquo2-D URANS vsexperiments of flow induced motions of two circular cylindersin tandemwith passive turbulence control for 30000leRele 105000rdquo Ocean Engineering vol 72 pp 429ndash440 2013

[23] A Abdelkefi M R Hajj and A H Nayfeh ldquoPhenomena andmodeling of piezoelectric energy harvesting from freely oscil-lating cylindersrdquo Nonlinear Dynamics An International Journalof Nonlinear Dynamics and Chaos in Engineering Systems vol70 no 2 pp 1377ndash1388 2012

[24] A Abdelkefi A H Nayfeh andM R Hajj ldquoModeling and anal-ysis of piezoaeroelastic energy harvestersrdquoNonlinear DynamicsAn International Journal of Nonlinear Dynamics and Chaos inEngineering Systems vol 67 no 2 pp 925ndash939 2012

[25] 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

[26] 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

[27] H L Dai A Abdelkefi and L Wang ldquoPiezoelectric energyharvesting from concurrent vortex-induced vibrations and baseexcitationsrdquo Nonlinear Dynamics vol 77 no 3 pp 967ndash9812014

[28] 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

[29] W P Robbins I Marusic D Morris et al ldquoWind-generatedelectrical energy using flexible piezoelectric mateialsrdquo in Pro-ceedings of theASME2006 InternationalMechanical EngineeringCongress and Exposition American Society of Mechanical Engi-neers vol 23 pp 581ndash590 2006

[30] H D Akaydin N Elvin and Y Andreopoulos ldquoWake of acylinder a paradigm for energy harvesting with piezoelectricmaterialsrdquo Experiments in Fluids vol 49 no 1 pp 291ndash3042010

[31] X Gao W-H Shih and W Y Shih ldquoFlow energy harvestingusing piezoelectric cantilevers with cylindrical extensionrdquo IEEETransactions on Industrial Electronics vol 60 no 3 pp 1116ndash1118 2013

18 Geofluids

[32] H D Akaydin N Elvin and Y Andreopoulos ldquoThe per-formance of a self-excited fluidic energy harvesterrdquo SmartMaterials Structures vol 21 no 2 Article ID 025007 2012

[33] P R Spalart and S R Allmaras ldquoA one-equation turbulencemodel for aerodynamic flowsrdquo La Recherche Aerospatiale vol439 no 1 pp 5ndash21 2003

[34] 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

Submit your manuscripts athttpswwwhindawicom

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2 Geofluids

device of a PVDF polymer to harvest marine energy whichwas placed in a water tank with the length of 2413mmthe width of 762mm and the thickness of 150 120583m Theresearch shows that when the flapping frequency of PVDFcloses to the vortex shedding frequency the energy collectionperformance will be improved the maximum voltage 3Vappears in the water flow velocity of 05ms In MarineRenewable Energy Laboratory (MRELab) at the Universityof Michigan Bernitsas et al [14 15] have presented a vortex-induced vibration for aquatic clean energy (VIVACE) toutilize the VIV phenomenon to generate power The latterstudies have been conducted in support of model tests forVIVACE converter which harnesses hydrokinetic energyenhancing flow induced motions (FIM) and particularlyVIV and various forms of galloping Lee and Bernitsas[16 17] built a devicesystem Vck to replace the physicaldampersprings of the VIVACE with virtual elements Thetesting was performed in the Low Turbulence Free SurfaceWater Channel of theUniversity ofMichigan at 40000 lt Re lt120000 and damping 0 lt 120585 lt 016 To continue and improvethe work of VIVACE which is used to convert hydrokineticenergy fromoceanriver currents to electricity Raghavan andBernitsas [18 19] conducted experiments in the regime rightbefore transition from laminar to turbulent flow Hysteresiswas not observed in the experiments because the parametersremain in the 2P domain Higher VIV amplitude of 21 to27 diameters was obtained and the range of synchronizationwas increased by increasing the damping in VIVACE Changet al [20] put a PTC device in the surface of the VIVACEconverter to enhance the viscous damping the results showthat proper PTC could enhance the conversion efficiency ofhydrokinetic energy to mechanical in VIVACE For solvingthe problem of VIV some excellent researches have alsobeen conducted in MRELab With the help of OpenFOAMsoftware Wu et al [21] have developed a CFD code to solvethe VIV problem of a single cylinder with PTC Ding etal [22] have also developed a CFD code in OpenFOAMto solve the VIV problem of multiple circular cylindersAbdelkefi et al [23 24] andMehmood et al [25] investigatedthe possibility of harvesting energy from elastically mountedcircular cylinders They used respectively a wake oscillatormodel and direct numerical simulations to determine thefluctuating lift force Abdelkefi et al [23] reported thatthe aerodynamic nonlinearity results in the presence ofhardening behavior Both research studies determined theeffects of the load resistance on the synchronization regionand level of the harvested power They demonstrated thatmaximum level of the harvested power can be associatedwith minimum values in the transverse displacement dueto the shunt damping effect To investigate the possibilityof harvesting energy from other flow induced vibrationsdifferent investigations have been performed Abdelkefi andNuhait [26] investigated the effects of cambered wing-basedpiezoaeroelastic energy harvesters on the flutter speed andthe performance of the harvester Dai et al [27] and Yan andAbdelkefi [28] investigated the potential of harvesting energyfrom a combination of base excitations and VIV or gallopingrespectively Robbins et al [29] proposed a device similarto the ldquoeelrdquo model consisting of piezoelectric film arrays

(single size 203 times 279 times 05mm) with different configurationsto collect wind energy generated by induced draft fan themaximum output power is up to 10mW Akaydin et al [30]proposed a structure (main size 30 times 16 times 02mm) whichcan separate the piezoelectric film from the bluff body inwhich the maximum output power is 4 120583W Gao et al [31]carried out an experiment model (unit piezoelectric elementsize is 58 times 10mm cylinder diameter is 291mm and cylinderlength is 36mm) in which the cylinder is connected withthe PZT cantilever beam and the PZT beam is fixed atone end in the wind tunnel Thus electrical charges wereaccompanied by the wind-induced vibration of the PZTcantilever beam The ldquolock-inrdquo phenomenon is observed inthe wind-induced vibration and a high voltage output isobtained in this region Akaydin et al [32] presented anenergy harvesting experiment in a wind tunnel in which thecylinder is connected with the free end of a cantilever beamwith aluminum shim device The results show that when thecircuit resistance is changed the natural frequency of thesystem will be changed also so that the maximum value ofthe electromechanical coupling damping can be obtainedThe above researches are mainly focused on the energycollection models of the cylindrical bluff body under vortex-induced vibration Studies on the vortex-induced vibrationandpiezoelectric energy harvesting of noncylindrical sectionsuch as square column and triangular column are relativelyless

It is very important to consider the different attack anglesin the analysis of square column vortex-induced vibrationbecause of the asymmetry The motion of the fluid at theedge of the square column is very irregular which willgreatly affect the vortex shedding near the column wall Thesquare column vortex shedding process is different fromthat of smooth circular cylinder there is a direct vortexshedding without a smooth transition region The boundarylayer separation point of the square column is fixed at thecorner of the leading edge without changing its position withthe varying of Reynolds number Therefore it is of greatpractical significance to study on the influence of differentattack angles of square column vortex-induced vibrationpiezoelectric energy harvesting (VIVPEH)

In this paper a model of square column VIVPEH ispresented which can be used to investigate the mechanismof different attack angles on the energy capturing efficiencyThe quasi steady state theory is adopted to describe the fluidsolid coupling process of vortex-induced vibration based onthe finite volume method coupled Gauss equation Finallythe performance of different attack angles on the VIVPEHis carried out with considering the flow field parameters thevibration characteristics and the energy collection parame-ters

2 Physical Model

The physical model of square column VIVPEH is shown inFigure 1 inwhich119880 stands for the inflowvelocity120572 stands forthe angle between the square column and the inflow velocity119872 is the mass of the column119870 is the elastic coefficient of the

Geofluids 3

R

U

PZT


R K C

Fy

FL

F

F

M

L0

U

D

x








120597119880119894120597119909119894 = 0120597119880119894120597119905 + 119880119895

120597119880119894120597119909119895 = minus1120588120597119901120597119909119894 +

120597120597119909119895 (2]119878119894119895 minus 11990610158401198951199061015840119894) (1)


119878119894119895 = 12 (120597119880119894120597119909119895 +

120597119880119895120597119909119894 ) (2)



120597]120597119905 + 119906119895 120597]120597120594119895 = 1198881198871] minus 1198881199081119891119908 (]119889)2

+ 1120590 120597120597120594119895 [(] + ]) 120597]120597120594119895] + 1198881198872120597]120597120594119894

sdot 120597]120597120594119895

(3)

4 Geofluids


120583119905 = 120588]119891V1 (4)




119872 + 119862 + 119870119884 = 119865119910 (5)


120596119899 = radic 119870119872119862 = 2120585119872120596119899

(6)



119872 + 119862 + 119870119884 minus 120579119881 = 119865119910 (7)

120579 + 119862119901 + 119881119877 = 0 (8)




119872 + 119862 + 119870119884 minus 120579119881 = 0 (9)



1198833119877119862119901 (10)


= 119861 (119877)119883 (11)

where

119883 = [1198831 1198832 1198833]119879

119861 (119877) =[[[[[[[


]]]]]]] (12)



119884 = 119884max sin (120596119899119905) (13)




minus 120596119899119877119862119901 sin (120596119899119905)) (14)

Geofluids 5





119875 (119905) = 1198812 (119905)119877 (15)



5 Numerical Results


119863Nor = 119863squ (sin120572 + cos120572) (16)





119880119903squ = 119880119891119899119863squ (17)



6 Geofluids

(a) 120572 = 0∘ (b) 120572 = 15∘ (c) 120572 = 30∘

(d) 120572 = 45∘ (e) 120572 = 60∘ (f) 120572 = 75∘

Square column

Top

y

x

OutletInlet

Bottom

(g)





Geofluids 7


119880 (ms) 119880119903squ120572 = 0∘ 120572 = 15∘ (120572 = 75∘) 120572 = 30∘ (120572 = 60∘) 120572 = 45∘003927 35 285784 256223 247525004488 4 326611 292826 282885005049 45 367437 329429 31824600561 5 408263 366032 353607005834 52 424594 380673 367751006059 54 440924 395315 381895006283 56 457255 409956 39604006732 6 489916 439239 424328007293 65 530742 475842 459689007854 7 571569 512445 49505008415 75 612395 549048 53041008976 8 653221 585652 565771009537 85 694048 622255 601132

Real Imaginary

00

05

10

15

20

25

30

35

Real

103 104 105 106 107102

R (ohms)

44

45

46

47

48

49

50

Imag

inar

y




00

01

02

03

04

5 83 74 6UrSqu

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

DSq

uY

Squ



8 Geofluids

minus001

000

001

6 8 10 124Time (s)

DSq

uY

Squ


0

200

400

600

800

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus006

minus004

minus002

000DSq

uY

Squ

002

004

006

18 21 24 27 3015Time (s)


0

1000

2000

3000M

agni

tude

4 5 6 7 8 9 103Frequency (Hz)






Geofluids 9

minus003

minus002

minus001

000DSq

uY

Squ

001

002

003

18 21 24 2715Time (s)


0

200

400

600

800

1000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000DSq

uY

Squ

005

010

015

18 21 2415Time (s)


0

2000

4000

6000

8000

10000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)









10 Geofluids

minus02

minus01

00DSq

uY

Squ

01

02

3 6 9 12 15 18 21 24 27 300Time (s)


0

2000

4000

6000

8000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus03

minus02

minus01

00

01

02

03

04

DSq

uY

Squ

3 6 9 12 15 18 21 24 27 300Time (s)


0

10000

20000

30000

40000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)





005049 01682 01682 01832 0183100561 01714 01714 01873 01872006278 01743 01742 01896 01896007293 01746 01746 01922 01922





Geofluids 11

minus0006

minus0004

minus0002

0000

0002

0004

0006D

Squ

YSq

u

1815 21 2724 3330Time (s)


0

100

200

300

400

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus03

minus02

minus01

00

01

02

03

DSq

uY

Squ

18 20 2216Time (s)


0

5000

10000

15000

20000

25000

30000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)







12 Geofluids

minus02

minus01

00

01

02

DSq

uY

Squ

5 10 15 20 25 30 35 400Time (s)


0

5000

10000

15000

20000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus02

00

02

04

DSq

uY

Squ

3 6 9 12 150Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

2000

4000

6000

8000

10000

12000

14000

16000

Mag

nitu

de






Geofluids 13

minus02

minus01

00

01

02

DSq

uY

Squ

18 21 24 27 30 3315Time (s)


0

500

1000

1500

2000

2500

3000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000

005

010

015

DSq

uY

Squ

18 21 24 27 3015Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

4700

9400

14100

18800M

agni

tude



06

08

10

12

85 6 743

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

f３ＫＯf

N３ＫＯ


UrSqu


14 Geofluids

minus0004

minus0003

minus0002

minus0001

0000

0001

0002

0003

0004

YD

minus15

minus10

minus05

00

05

10

15

CL

158154 156 160150 152

Time (s)


minus006

minus004

minus002

000

002

004

006

YD

158154 156 160150 152

Time (s)

minus15

minus10

minus05

00

05

10

15

CL



minus0004

minus0002

0000

0002

0004

YD

152 154 160158156150

Time (s)

minus10

minus05

00

05

10

CL


minus015

minus010

minus005

000

005

010

015YD

152 154 156 158 160150

Time (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

CL



minus0002

0000

0002

YD

151 158157156 159150 154153152 160155

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus0004

0004


minus04

minus03

minus02

minus01

00

01

02

03

04

YD

151 152 159154 158156 157 160150 153 155

Time (s)

minus25

minus20

minus15

minus10

minus05

00

05

10

15C

L



Geofluids 15

minus0006

minus0004

minus0002

0000

0002

0004

0006YD

151 152 153 154 155 156 157 158 159 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL


151 155153 157 159156152 158154 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus02

minus01

00

01

02

YD



V1

V1

V2

V2

S1


V3

V4

S2

V3

V4



tT３ＫＯ0 = 0 tT３ＫＯ0 = 0249

tT３ＫＯ0 = 0502 tT３ＫＯ0 = 075



tT３ＫＯ15 = 0 tT３ＫＯ15 = 025

tT３ＫＯ15 = 0497 tT３ＫＯ15 = 0752



16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions







effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)


00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (






Acknowledgments


Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

Geofluids 3

R

U

PZT


R K C

Fy

FL

F

F

M

L0

U

D

x








120597119880119894120597119909119894 = 0120597119880119894120597119905 + 119880119895

120597119880119894120597119909119895 = minus1120588120597119901120597119909119894 +

120597120597119909119895 (2]119878119894119895 minus 11990610158401198951199061015840119894) (1)


119878119894119895 = 12 (120597119880119894120597119909119895 +

120597119880119895120597119909119894 ) (2)



120597]120597119905 + 119906119895 120597]120597120594119895 = 1198881198871] minus 1198881199081119891119908 (]119889)2

+ 1120590 120597120597120594119895 [(] + ]) 120597]120597120594119895] + 1198881198872120597]120597120594119894

sdot 120597]120597120594119895

(3)

4 Geofluids


120583119905 = 120588]119891V1 (4)




119872 + 119862 + 119870119884 = 119865119910 (5)


120596119899 = radic 119870119872119862 = 2120585119872120596119899

(6)



119872 + 119862 + 119870119884 minus 120579119881 = 119865119910 (7)

120579 + 119862119901 + 119881119877 = 0 (8)




119872 + 119862 + 119870119884 minus 120579119881 = 0 (9)



1198833119877119862119901 (10)


= 119861 (119877)119883 (11)

where

119883 = [1198831 1198832 1198833]119879

119861 (119877) =[[[[[[[


]]]]]]] (12)



119884 = 119884max sin (120596119899119905) (13)




minus 120596119899119877119862119901 sin (120596119899119905)) (14)

Geofluids 5





119875 (119905) = 1198812 (119905)119877 (15)



5 Numerical Results


119863Nor = 119863squ (sin120572 + cos120572) (16)





119880119903squ = 119880119891119899119863squ (17)



6 Geofluids

(a) 120572 = 0∘ (b) 120572 = 15∘ (c) 120572 = 30∘

(d) 120572 = 45∘ (e) 120572 = 60∘ (f) 120572 = 75∘

Square column

Top

y

x

OutletInlet

Bottom

(g)





Geofluids 7


119880 (ms) 119880119903squ120572 = 0∘ 120572 = 15∘ (120572 = 75∘) 120572 = 30∘ (120572 = 60∘) 120572 = 45∘003927 35 285784 256223 247525004488 4 326611 292826 282885005049 45 367437 329429 31824600561 5 408263 366032 353607005834 52 424594 380673 367751006059 54 440924 395315 381895006283 56 457255 409956 39604006732 6 489916 439239 424328007293 65 530742 475842 459689007854 7 571569 512445 49505008415 75 612395 549048 53041008976 8 653221 585652 565771009537 85 694048 622255 601132

Real Imaginary

00

05

10

15

20

25

30

35

Real

103 104 105 106 107102

R (ohms)

44

45

46

47

48

49

50

Imag

inar

y




00

01

02

03

04

5 83 74 6UrSqu

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

DSq

uY

Squ



8 Geofluids

minus001

000

001

6 8 10 124Time (s)

DSq

uY

Squ


0

200

400

600

800

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus006

minus004

minus002

000DSq

uY

Squ

002

004

006

18 21 24 27 3015Time (s)


0

1000

2000

3000M

agni

tude

4 5 6 7 8 9 103Frequency (Hz)






Geofluids 9

minus003

minus002

minus001

000DSq

uY

Squ

001

002

003

18 21 24 2715Time (s)


0

200

400

600

800

1000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000DSq

uY

Squ

005

010

015

18 21 2415Time (s)


0

2000

4000

6000

8000

10000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)









10 Geofluids

minus02

minus01

00DSq

uY

Squ

01

02

3 6 9 12 15 18 21 24 27 300Time (s)


0

2000

4000

6000

8000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus03

minus02

minus01

00

01

02

03

04

DSq

uY

Squ

3 6 9 12 15 18 21 24 27 300Time (s)


0

10000

20000

30000

40000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)





005049 01682 01682 01832 0183100561 01714 01714 01873 01872006278 01743 01742 01896 01896007293 01746 01746 01922 01922





Geofluids 11

minus0006

minus0004

minus0002

0000

0002

0004

0006D

Squ

YSq

u

1815 21 2724 3330Time (s)


0

100

200

300

400

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus03

minus02

minus01

00

01

02

03

DSq

uY

Squ

18 20 2216Time (s)


0

5000

10000

15000

20000

25000

30000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)







12 Geofluids

minus02

minus01

00

01

02

DSq

uY

Squ

5 10 15 20 25 30 35 400Time (s)


0

5000

10000

15000

20000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus02

00

02

04

DSq

uY

Squ

3 6 9 12 150Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

2000

4000

6000

8000

10000

12000

14000

16000

Mag

nitu

de






Geofluids 13

minus02

minus01

00

01

02

DSq

uY

Squ

18 21 24 27 30 3315Time (s)


0

500

1000

1500

2000

2500

3000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000

005

010

015

DSq

uY

Squ

18 21 24 27 3015Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

4700

9400

14100

18800M

agni

tude



06

08

10

12

85 6 743

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

f３ＫＯf

N３ＫＯ


UrSqu


14 Geofluids

minus0004

minus0003

minus0002

minus0001

0000

0001

0002

0003

0004

YD

minus15

minus10

minus05

00

05

10

15

CL

158154 156 160150 152

Time (s)


minus006

minus004

minus002

000

002

004

006

YD

158154 156 160150 152

Time (s)

minus15

minus10

minus05

00

05

10

15

CL



minus0004

minus0002

0000

0002

0004

YD

152 154 160158156150

Time (s)

minus10

minus05

00

05

10

CL


minus015

minus010

minus005

000

005

010

015YD

152 154 156 158 160150

Time (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

CL



minus0002

0000

0002

YD

151 158157156 159150 154153152 160155

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus0004

0004


minus04

minus03

minus02

minus01

00

01

02

03

04

YD

151 152 159154 158156 157 160150 153 155

Time (s)

minus25

minus20

minus15

minus10

minus05

00

05

10

15C

L



Geofluids 15

minus0006

minus0004

minus0002

0000

0002

0004

0006YD

151 152 153 154 155 156 157 158 159 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL


151 155153 157 159156152 158154 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus02

minus01

00

01

02

YD



V1

V1

V2

V2

S1


V3

V4

S2

V3

V4



tT３ＫＯ0 = 0 tT３ＫＯ0 = 0249

tT３ＫＯ0 = 0502 tT３ＫＯ0 = 075



tT３ＫＯ15 = 0 tT３ＫＯ15 = 025

tT３ＫＯ15 = 0497 tT３ＫＯ15 = 0752



16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions







effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)


00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (






Acknowledgments


Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

4 Geofluids


120583119905 = 120588]119891V1 (4)




119872 + 119862 + 119870119884 = 119865119910 (5)


120596119899 = radic 119870119872119862 = 2120585119872120596119899

(6)



119872 + 119862 + 119870119884 minus 120579119881 = 119865119910 (7)

120579 + 119862119901 + 119881119877 = 0 (8)




119872 + 119862 + 119870119884 minus 120579119881 = 0 (9)



1198833119877119862119901 (10)


= 119861 (119877)119883 (11)

where

119883 = [1198831 1198832 1198833]119879

119861 (119877) =[[[[[[[


]]]]]]] (12)



119884 = 119884max sin (120596119899119905) (13)




minus 120596119899119877119862119901 sin (120596119899119905)) (14)

Geofluids 5





119875 (119905) = 1198812 (119905)119877 (15)



5 Numerical Results


119863Nor = 119863squ (sin120572 + cos120572) (16)





119880119903squ = 119880119891119899119863squ (17)



6 Geofluids

(a) 120572 = 0∘ (b) 120572 = 15∘ (c) 120572 = 30∘

(d) 120572 = 45∘ (e) 120572 = 60∘ (f) 120572 = 75∘

Square column

Top

y

x

OutletInlet

Bottom

(g)





Geofluids 7


119880 (ms) 119880119903squ120572 = 0∘ 120572 = 15∘ (120572 = 75∘) 120572 = 30∘ (120572 = 60∘) 120572 = 45∘003927 35 285784 256223 247525004488 4 326611 292826 282885005049 45 367437 329429 31824600561 5 408263 366032 353607005834 52 424594 380673 367751006059 54 440924 395315 381895006283 56 457255 409956 39604006732 6 489916 439239 424328007293 65 530742 475842 459689007854 7 571569 512445 49505008415 75 612395 549048 53041008976 8 653221 585652 565771009537 85 694048 622255 601132

Real Imaginary

00

05

10

15

20

25

30

35

Real

103 104 105 106 107102

R (ohms)

44

45

46

47

48

49

50

Imag

inar

y




00

01

02

03

04

5 83 74 6UrSqu

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

DSq

uY

Squ



8 Geofluids

minus001

000

001

6 8 10 124Time (s)

DSq

uY

Squ


0

200

400

600

800

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus006

minus004

minus002

000DSq

uY

Squ

002

004

006

18 21 24 27 3015Time (s)


0

1000

2000

3000M

agni

tude

4 5 6 7 8 9 103Frequency (Hz)






Geofluids 9

minus003

minus002

minus001

000DSq

uY

Squ

001

002

003

18 21 24 2715Time (s)


0

200

400

600

800

1000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000DSq

uY

Squ

005

010

015

18 21 2415Time (s)


0

2000

4000

6000

8000

10000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)









10 Geofluids

minus02

minus01

00DSq

uY

Squ

01

02

3 6 9 12 15 18 21 24 27 300Time (s)


0

2000

4000

6000

8000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus03

minus02

minus01

00

01

02

03

04

DSq

uY

Squ

3 6 9 12 15 18 21 24 27 300Time (s)


0

10000

20000

30000

40000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)





005049 01682 01682 01832 0183100561 01714 01714 01873 01872006278 01743 01742 01896 01896007293 01746 01746 01922 01922





Geofluids 11

minus0006

minus0004

minus0002

0000

0002

0004

0006D

Squ

YSq

u

1815 21 2724 3330Time (s)


0

100

200

300

400

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus03

minus02

minus01

00

01

02

03

DSq

uY

Squ

18 20 2216Time (s)


0

5000

10000

15000

20000

25000

30000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)







12 Geofluids

minus02

minus01

00

01

02

DSq

uY

Squ

5 10 15 20 25 30 35 400Time (s)


0

5000

10000

15000

20000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus02

00

02

04

DSq

uY

Squ

3 6 9 12 150Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

2000

4000

6000

8000

10000

12000

14000

16000

Mag

nitu

de






Geofluids 13

minus02

minus01

00

01

02

DSq

uY

Squ

18 21 24 27 30 3315Time (s)


0

500

1000

1500

2000

2500

3000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000

005

010

015

DSq

uY

Squ

18 21 24 27 3015Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

4700

9400

14100

18800M

agni

tude



06

08

10

12

85 6 743

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

f３ＫＯf

N３ＫＯ


UrSqu


14 Geofluids

minus0004

minus0003

minus0002

minus0001

0000

0001

0002

0003

0004

YD

minus15

minus10

minus05

00

05

10

15

CL

158154 156 160150 152

Time (s)


minus006

minus004

minus002

000

002

004

006

YD

158154 156 160150 152

Time (s)

minus15

minus10

minus05

00

05

10

15

CL



minus0004

minus0002

0000

0002

0004

YD

152 154 160158156150

Time (s)

minus10

minus05

00

05

10

CL


minus015

minus010

minus005

000

005

010

015YD

152 154 156 158 160150

Time (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

CL



minus0002

0000

0002

YD

151 158157156 159150 154153152 160155

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus0004

0004


minus04

minus03

minus02

minus01

00

01

02

03

04

YD

151 152 159154 158156 157 160150 153 155

Time (s)

minus25

minus20

minus15

minus10

minus05

00

05

10

15C

L



Geofluids 15

minus0006

minus0004

minus0002

0000

0002

0004

0006YD

151 152 153 154 155 156 157 158 159 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL


151 155153 157 159156152 158154 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus02

minus01

00

01

02

YD



V1

V1

V2

V2

S1


V3

V4

S2

V3

V4



tT３ＫＯ0 = 0 tT３ＫＯ0 = 0249

tT３ＫＯ0 = 0502 tT３ＫＯ0 = 075



tT３ＫＯ15 = 0 tT３ＫＯ15 = 025

tT３ＫＯ15 = 0497 tT３ＫＯ15 = 0752



16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions







effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)


00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (






Acknowledgments


Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

Geofluids 5





119875 (119905) = 1198812 (119905)119877 (15)



5 Numerical Results


119863Nor = 119863squ (sin120572 + cos120572) (16)





119880119903squ = 119880119891119899119863squ (17)



6 Geofluids

(a) 120572 = 0∘ (b) 120572 = 15∘ (c) 120572 = 30∘

(d) 120572 = 45∘ (e) 120572 = 60∘ (f) 120572 = 75∘

Square column

Top

y

x

OutletInlet

Bottom

(g)





Geofluids 7


119880 (ms) 119880119903squ120572 = 0∘ 120572 = 15∘ (120572 = 75∘) 120572 = 30∘ (120572 = 60∘) 120572 = 45∘003927 35 285784 256223 247525004488 4 326611 292826 282885005049 45 367437 329429 31824600561 5 408263 366032 353607005834 52 424594 380673 367751006059 54 440924 395315 381895006283 56 457255 409956 39604006732 6 489916 439239 424328007293 65 530742 475842 459689007854 7 571569 512445 49505008415 75 612395 549048 53041008976 8 653221 585652 565771009537 85 694048 622255 601132

Real Imaginary

00

05

10

15

20

25

30

35

Real

103 104 105 106 107102

R (ohms)

44

45

46

47

48

49

50

Imag

inar

y




00

01

02

03

04

5 83 74 6UrSqu

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

DSq

uY

Squ



8 Geofluids

minus001

000

001

6 8 10 124Time (s)

DSq

uY

Squ


0

200

400

600

800

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus006

minus004

minus002

000DSq

uY

Squ

002

004

006

18 21 24 27 3015Time (s)


0

1000

2000

3000M

agni

tude

4 5 6 7 8 9 103Frequency (Hz)






Geofluids 9

minus003

minus002

minus001

000DSq

uY

Squ

001

002

003

18 21 24 2715Time (s)


0

200

400

600

800

1000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000DSq

uY

Squ

005

010

015

18 21 2415Time (s)


0

2000

4000

6000

8000

10000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)









10 Geofluids

minus02

minus01

00DSq

uY

Squ

01

02

3 6 9 12 15 18 21 24 27 300Time (s)


0

2000

4000

6000

8000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus03

minus02

minus01

00

01

02

03

04

DSq

uY

Squ

3 6 9 12 15 18 21 24 27 300Time (s)


0

10000

20000

30000

40000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)





005049 01682 01682 01832 0183100561 01714 01714 01873 01872006278 01743 01742 01896 01896007293 01746 01746 01922 01922





Geofluids 11

minus0006

minus0004

minus0002

0000

0002

0004

0006D

Squ

YSq

u

1815 21 2724 3330Time (s)


0

100

200

300

400

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus03

minus02

minus01

00

01

02

03

DSq

uY

Squ

18 20 2216Time (s)


0

5000

10000

15000

20000

25000

30000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)







12 Geofluids

minus02

minus01

00

01

02

DSq

uY

Squ

5 10 15 20 25 30 35 400Time (s)


0

5000

10000

15000

20000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus02

00

02

04

DSq

uY

Squ

3 6 9 12 150Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

2000

4000

6000

8000

10000

12000

14000

16000

Mag

nitu

de






Geofluids 13

minus02

minus01

00

01

02

DSq

uY

Squ

18 21 24 27 30 3315Time (s)


0

500

1000

1500

2000

2500

3000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000

005

010

015

DSq

uY

Squ

18 21 24 27 3015Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

4700

9400

14100

18800M

agni

tude



06

08

10

12

85 6 743

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

f３ＫＯf

N３ＫＯ


UrSqu


14 Geofluids

minus0004

minus0003

minus0002

minus0001

0000

0001

0002

0003

0004

YD

minus15

minus10

minus05

00

05

10

15

CL

158154 156 160150 152

Time (s)


minus006

minus004

minus002

000

002

004

006

YD

158154 156 160150 152

Time (s)

minus15

minus10

minus05

00

05

10

15

CL



minus0004

minus0002

0000

0002

0004

YD

152 154 160158156150

Time (s)

minus10

minus05

00

05

10

CL


minus015

minus010

minus005

000

005

010

015YD

152 154 156 158 160150

Time (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

CL



minus0002

0000

0002

YD

151 158157156 159150 154153152 160155

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus0004

0004


minus04

minus03

minus02

minus01

00

01

02

03

04

YD

151 152 159154 158156 157 160150 153 155

Time (s)

minus25

minus20

minus15

minus10

minus05

00

05

10

15C

L



Geofluids 15

minus0006

minus0004

minus0002

0000

0002

0004

0006YD

151 152 153 154 155 156 157 158 159 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL


151 155153 157 159156152 158154 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus02

minus01

00

01

02

YD



V1

V1

V2

V2

S1


V3

V4

S2

V3

V4



tT３ＫＯ0 = 0 tT３ＫＯ0 = 0249

tT３ＫＯ0 = 0502 tT３ＫＯ0 = 075



tT３ＫＯ15 = 0 tT３ＫＯ15 = 025

tT３ＫＯ15 = 0497 tT３ＫＯ15 = 0752



16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions







effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)


00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (






Acknowledgments


Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

6 Geofluids

(a) 120572 = 0∘ (b) 120572 = 15∘ (c) 120572 = 30∘

(d) 120572 = 45∘ (e) 120572 = 60∘ (f) 120572 = 75∘

Square column

Top

y

x

OutletInlet

Bottom

(g)





Geofluids 7


119880 (ms) 119880119903squ120572 = 0∘ 120572 = 15∘ (120572 = 75∘) 120572 = 30∘ (120572 = 60∘) 120572 = 45∘003927 35 285784 256223 247525004488 4 326611 292826 282885005049 45 367437 329429 31824600561 5 408263 366032 353607005834 52 424594 380673 367751006059 54 440924 395315 381895006283 56 457255 409956 39604006732 6 489916 439239 424328007293 65 530742 475842 459689007854 7 571569 512445 49505008415 75 612395 549048 53041008976 8 653221 585652 565771009537 85 694048 622255 601132

Real Imaginary

00

05

10

15

20

25

30

35

Real

103 104 105 106 107102

R (ohms)

44

45

46

47

48

49

50

Imag

inar

y




00

01

02

03

04

5 83 74 6UrSqu

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

DSq

uY

Squ



8 Geofluids

minus001

000

001

6 8 10 124Time (s)

DSq

uY

Squ


0

200

400

600

800

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus006

minus004

minus002

000DSq

uY

Squ

002

004

006

18 21 24 27 3015Time (s)


0

1000

2000

3000M

agni

tude

4 5 6 7 8 9 103Frequency (Hz)






Geofluids 9

minus003

minus002

minus001

000DSq

uY

Squ

001

002

003

18 21 24 2715Time (s)


0

200

400

600

800

1000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000DSq

uY

Squ

005

010

015

18 21 2415Time (s)


0

2000

4000

6000

8000

10000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)









10 Geofluids

minus02

minus01

00DSq

uY

Squ

01

02

3 6 9 12 15 18 21 24 27 300Time (s)


0

2000

4000

6000

8000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus03

minus02

minus01

00

01

02

03

04

DSq

uY

Squ

3 6 9 12 15 18 21 24 27 300Time (s)


0

10000

20000

30000

40000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)





005049 01682 01682 01832 0183100561 01714 01714 01873 01872006278 01743 01742 01896 01896007293 01746 01746 01922 01922





Geofluids 11

minus0006

minus0004

minus0002

0000

0002

0004

0006D

Squ

YSq

u

1815 21 2724 3330Time (s)


0

100

200

300

400

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus03

minus02

minus01

00

01

02

03

DSq

uY

Squ

18 20 2216Time (s)


0

5000

10000

15000

20000

25000

30000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)







12 Geofluids

minus02

minus01

00

01

02

DSq

uY

Squ

5 10 15 20 25 30 35 400Time (s)


0

5000

10000

15000

20000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus02

00

02

04

DSq

uY

Squ

3 6 9 12 150Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

2000

4000

6000

8000

10000

12000

14000

16000

Mag

nitu

de






Geofluids 13

minus02

minus01

00

01

02

DSq

uY

Squ

18 21 24 27 30 3315Time (s)


0

500

1000

1500

2000

2500

3000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000

005

010

015

DSq

uY

Squ

18 21 24 27 3015Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

4700

9400

14100

18800M

agni

tude



06

08

10

12

85 6 743

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

f３ＫＯf

N３ＫＯ


UrSqu


14 Geofluids

minus0004

minus0003

minus0002

minus0001

0000

0001

0002

0003

0004

YD

minus15

minus10

minus05

00

05

10

15

CL

158154 156 160150 152

Time (s)


minus006

minus004

minus002

000

002

004

006

YD

158154 156 160150 152

Time (s)

minus15

minus10

minus05

00

05

10

15

CL



minus0004

minus0002

0000

0002

0004

YD

152 154 160158156150

Time (s)

minus10

minus05

00

05

10

CL


minus015

minus010

minus005

000

005

010

015YD

152 154 156 158 160150

Time (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

CL



minus0002

0000

0002

YD

151 158157156 159150 154153152 160155

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus0004

0004


minus04

minus03

minus02

minus01

00

01

02

03

04

YD

151 152 159154 158156 157 160150 153 155

Time (s)

minus25

minus20

minus15

minus10

minus05

00

05

10

15C

L



Geofluids 15

minus0006

minus0004

minus0002

0000

0002

0004

0006YD

151 152 153 154 155 156 157 158 159 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL


151 155153 157 159156152 158154 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus02

minus01

00

01

02

YD



V1

V1

V2

V2

S1


V3

V4

S2

V3

V4



tT３ＫＯ0 = 0 tT３ＫＯ0 = 0249

tT３ＫＯ0 = 0502 tT３ＫＯ0 = 075



tT３ＫＯ15 = 0 tT３ＫＯ15 = 025

tT３ＫＯ15 = 0497 tT３ＫＯ15 = 0752



16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions







effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)


00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (






Acknowledgments


Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

Geofluids 7


119880 (ms) 119880119903squ120572 = 0∘ 120572 = 15∘ (120572 = 75∘) 120572 = 30∘ (120572 = 60∘) 120572 = 45∘003927 35 285784 256223 247525004488 4 326611 292826 282885005049 45 367437 329429 31824600561 5 408263 366032 353607005834 52 424594 380673 367751006059 54 440924 395315 381895006283 56 457255 409956 39604006732 6 489916 439239 424328007293 65 530742 475842 459689007854 7 571569 512445 49505008415 75 612395 549048 53041008976 8 653221 585652 565771009537 85 694048 622255 601132

Real Imaginary

00

05

10

15

20

25

30

35

Real

103 104 105 106 107102

R (ohms)

44

45

46

47

48

49

50

Imag

inar

y




00

01

02

03

04

5 83 74 6UrSqu

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

DSq

uY

Squ



8 Geofluids

minus001

000

001

6 8 10 124Time (s)

DSq

uY

Squ


0

200

400

600

800

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus006

minus004

minus002

000DSq

uY

Squ

002

004

006

18 21 24 27 3015Time (s)


0

1000

2000

3000M

agni

tude

4 5 6 7 8 9 103Frequency (Hz)






Geofluids 9

minus003

minus002

minus001

000DSq

uY

Squ

001

002

003

18 21 24 2715Time (s)


0

200

400

600

800

1000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000DSq

uY

Squ

005

010

015

18 21 2415Time (s)


0

2000

4000

6000

8000

10000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)









10 Geofluids

minus02

minus01

00DSq

uY

Squ

01

02

3 6 9 12 15 18 21 24 27 300Time (s)


0

2000

4000

6000

8000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus03

minus02

minus01

00

01

02

03

04

DSq

uY

Squ

3 6 9 12 15 18 21 24 27 300Time (s)


0

10000

20000

30000

40000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)





005049 01682 01682 01832 0183100561 01714 01714 01873 01872006278 01743 01742 01896 01896007293 01746 01746 01922 01922





Geofluids 11

minus0006

minus0004

minus0002

0000

0002

0004

0006D

Squ

YSq

u

1815 21 2724 3330Time (s)


0

100

200

300

400

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus03

minus02

minus01

00

01

02

03

DSq

uY

Squ

18 20 2216Time (s)


0

5000

10000

15000

20000

25000

30000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)







12 Geofluids

minus02

minus01

00

01

02

DSq

uY

Squ

5 10 15 20 25 30 35 400Time (s)


0

5000

10000

15000

20000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus02

00

02

04

DSq

uY

Squ

3 6 9 12 150Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

2000

4000

6000

8000

10000

12000

14000

16000

Mag

nitu

de






Geofluids 13

minus02

minus01

00

01

02

DSq

uY

Squ

18 21 24 27 30 3315Time (s)


0

500

1000

1500

2000

2500

3000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000

005

010

015

DSq

uY

Squ

18 21 24 27 3015Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

4700

9400

14100

18800M

agni

tude



06

08

10

12

85 6 743

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

f３ＫＯf

N３ＫＯ


UrSqu


14 Geofluids

minus0004

minus0003

minus0002

minus0001

0000

0001

0002

0003

0004

YD

minus15

minus10

minus05

00

05

10

15

CL

158154 156 160150 152

Time (s)


minus006

minus004

minus002

000

002

004

006

YD

158154 156 160150 152

Time (s)

minus15

minus10

minus05

00

05

10

15

CL



minus0004

minus0002

0000

0002

0004

YD

152 154 160158156150

Time (s)

minus10

minus05

00

05

10

CL


minus015

minus010

minus005

000

005

010

015YD

152 154 156 158 160150

Time (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

CL



minus0002

0000

0002

YD

151 158157156 159150 154153152 160155

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus0004

0004


minus04

minus03

minus02

minus01

00

01

02

03

04

YD

151 152 159154 158156 157 160150 153 155

Time (s)

minus25

minus20

minus15

minus10

minus05

00

05

10

15C

L



Geofluids 15

minus0006

minus0004

minus0002

0000

0002

0004

0006YD

151 152 153 154 155 156 157 158 159 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL


151 155153 157 159156152 158154 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus02

minus01

00

01

02

YD



V1

V1

V2

V2

S1


V3

V4

S2

V3

V4



tT３ＫＯ0 = 0 tT３ＫＯ0 = 0249

tT３ＫＯ0 = 0502 tT３ＫＯ0 = 075



tT３ＫＯ15 = 0 tT３ＫＯ15 = 025

tT３ＫＯ15 = 0497 tT３ＫＯ15 = 0752



16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions







effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)


00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (






Acknowledgments


Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

8 Geofluids

minus001

000

001

6 8 10 124Time (s)

DSq

uY

Squ


0

200

400

600

800

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus006

minus004

minus002

000DSq

uY

Squ

002

004

006

18 21 24 27 3015Time (s)


0

1000

2000

3000M

agni

tude

4 5 6 7 8 9 103Frequency (Hz)






Geofluids 9

minus003

minus002

minus001

000DSq

uY

Squ

001

002

003

18 21 24 2715Time (s)


0

200

400

600

800

1000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000DSq

uY

Squ

005

010

015

18 21 2415Time (s)


0

2000

4000

6000

8000

10000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)









10 Geofluids

minus02

minus01

00DSq

uY

Squ

01

02

3 6 9 12 15 18 21 24 27 300Time (s)


0

2000

4000

6000

8000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus03

minus02

minus01

00

01

02

03

04

DSq

uY

Squ

3 6 9 12 15 18 21 24 27 300Time (s)


0

10000

20000

30000

40000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)





005049 01682 01682 01832 0183100561 01714 01714 01873 01872006278 01743 01742 01896 01896007293 01746 01746 01922 01922





Geofluids 11

minus0006

minus0004

minus0002

0000

0002

0004

0006D

Squ

YSq

u

1815 21 2724 3330Time (s)


0

100

200

300

400

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus03

minus02

minus01

00

01

02

03

DSq

uY

Squ

18 20 2216Time (s)


0

5000

10000

15000

20000

25000

30000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)







12 Geofluids

minus02

minus01

00

01

02

DSq

uY

Squ

5 10 15 20 25 30 35 400Time (s)


0

5000

10000

15000

20000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus02

00

02

04

DSq

uY

Squ

3 6 9 12 150Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

2000

4000

6000

8000

10000

12000

14000

16000

Mag

nitu

de






Geofluids 13

minus02

minus01

00

01

02

DSq

uY

Squ

18 21 24 27 30 3315Time (s)


0

500

1000

1500

2000

2500

3000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000

005

010

015

DSq

uY

Squ

18 21 24 27 3015Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

4700

9400

14100

18800M

agni

tude



06

08

10

12

85 6 743

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

f３ＫＯf

N３ＫＯ


UrSqu


14 Geofluids

minus0004

minus0003

minus0002

minus0001

0000

0001

0002

0003

0004

YD

minus15

minus10

minus05

00

05

10

15

CL

158154 156 160150 152

Time (s)


minus006

minus004

minus002

000

002

004

006

YD

158154 156 160150 152

Time (s)

minus15

minus10

minus05

00

05

10

15

CL



minus0004

minus0002

0000

0002

0004

YD

152 154 160158156150

Time (s)

minus10

minus05

00

05

10

CL


minus015

minus010

minus005

000

005

010

015YD

152 154 156 158 160150

Time (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

CL



minus0002

0000

0002

YD

151 158157156 159150 154153152 160155

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus0004

0004


minus04

minus03

minus02

minus01

00

01

02

03

04

YD

151 152 159154 158156 157 160150 153 155

Time (s)

minus25

minus20

minus15

minus10

minus05

00

05

10

15C

L



Geofluids 15

minus0006

minus0004

minus0002

0000

0002

0004

0006YD

151 152 153 154 155 156 157 158 159 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL


151 155153 157 159156152 158154 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus02

minus01

00

01

02

YD



V1

V1

V2

V2

S1


V3

V4

S2

V3

V4



tT３ＫＯ0 = 0 tT３ＫＯ0 = 0249

tT３ＫＯ0 = 0502 tT３ＫＯ0 = 075



tT３ＫＯ15 = 0 tT３ＫＯ15 = 025

tT３ＫＯ15 = 0497 tT３ＫＯ15 = 0752



16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions







effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)


00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (






Acknowledgments


Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

Geofluids 9

minus003

minus002

minus001

000DSq

uY

Squ

001

002

003

18 21 24 2715Time (s)


0

200

400

600

800

1000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000DSq

uY

Squ

005

010

015

18 21 2415Time (s)


0

2000

4000

6000

8000

10000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)









10 Geofluids

minus02

minus01

00DSq

uY

Squ

01

02

3 6 9 12 15 18 21 24 27 300Time (s)


0

2000

4000

6000

8000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus03

minus02

minus01

00

01

02

03

04

DSq

uY

Squ

3 6 9 12 15 18 21 24 27 300Time (s)


0

10000

20000

30000

40000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)





005049 01682 01682 01832 0183100561 01714 01714 01873 01872006278 01743 01742 01896 01896007293 01746 01746 01922 01922





Geofluids 11

minus0006

minus0004

minus0002

0000

0002

0004

0006D

Squ

YSq

u

1815 21 2724 3330Time (s)


0

100

200

300

400

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus03

minus02

minus01

00

01

02

03

DSq

uY

Squ

18 20 2216Time (s)


0

5000

10000

15000

20000

25000

30000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)







12 Geofluids

minus02

minus01

00

01

02

DSq

uY

Squ

5 10 15 20 25 30 35 400Time (s)


0

5000

10000

15000

20000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus02

00

02

04

DSq

uY

Squ

3 6 9 12 150Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

2000

4000

6000

8000

10000

12000

14000

16000

Mag

nitu

de






Geofluids 13

minus02

minus01

00

01

02

DSq

uY

Squ

18 21 24 27 30 3315Time (s)


0

500

1000

1500

2000

2500

3000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000

005

010

015

DSq

uY

Squ

18 21 24 27 3015Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

4700

9400

14100

18800M

agni

tude



06

08

10

12

85 6 743

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

f３ＫＯf

N３ＫＯ


UrSqu


14 Geofluids

minus0004

minus0003

minus0002

minus0001

0000

0001

0002

0003

0004

YD

minus15

minus10

minus05

00

05

10

15

CL

158154 156 160150 152

Time (s)


minus006

minus004

minus002

000

002

004

006

YD

158154 156 160150 152

Time (s)

minus15

minus10

minus05

00

05

10

15

CL



minus0004

minus0002

0000

0002

0004

YD

152 154 160158156150

Time (s)

minus10

minus05

00

05

10

CL


minus015

minus010

minus005

000

005

010

015YD

152 154 156 158 160150

Time (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

CL



minus0002

0000

0002

YD

151 158157156 159150 154153152 160155

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus0004

0004


minus04

minus03

minus02

minus01

00

01

02

03

04

YD

151 152 159154 158156 157 160150 153 155

Time (s)

minus25

minus20

minus15

minus10

minus05

00

05

10

15C

L



Geofluids 15

minus0006

minus0004

minus0002

0000

0002

0004

0006YD

151 152 153 154 155 156 157 158 159 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL


151 155153 157 159156152 158154 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus02

minus01

00

01

02

YD



V1

V1

V2

V2

S1


V3

V4

S2

V3

V4



tT３ＫＯ0 = 0 tT３ＫＯ0 = 0249

tT３ＫＯ0 = 0502 tT３ＫＯ0 = 075



tT３ＫＯ15 = 0 tT３ＫＯ15 = 025

tT３ＫＯ15 = 0497 tT３ＫＯ15 = 0752



16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions







effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)


00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (






Acknowledgments


Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

10 Geofluids

minus02

minus01

00DSq

uY

Squ

01

02

3 6 9 12 15 18 21 24 27 300Time (s)


0

2000

4000

6000

8000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus03

minus02

minus01

00

01

02

03

04

DSq

uY

Squ

3 6 9 12 15 18 21 24 27 300Time (s)


0

10000

20000

30000

40000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)





005049 01682 01682 01832 0183100561 01714 01714 01873 01872006278 01743 01742 01896 01896007293 01746 01746 01922 01922





Geofluids 11

minus0006

minus0004

minus0002

0000

0002

0004

0006D

Squ

YSq

u

1815 21 2724 3330Time (s)


0

100

200

300

400

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus03

minus02

minus01

00

01

02

03

DSq

uY

Squ

18 20 2216Time (s)


0

5000

10000

15000

20000

25000

30000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)







12 Geofluids

minus02

minus01

00

01

02

DSq

uY

Squ

5 10 15 20 25 30 35 400Time (s)


0

5000

10000

15000

20000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus02

00

02

04

DSq

uY

Squ

3 6 9 12 150Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

2000

4000

6000

8000

10000

12000

14000

16000

Mag

nitu

de






Geofluids 13

minus02

minus01

00

01

02

DSq

uY

Squ

18 21 24 27 30 3315Time (s)


0

500

1000

1500

2000

2500

3000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000

005

010

015

DSq

uY

Squ

18 21 24 27 3015Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

4700

9400

14100

18800M

agni

tude



06

08

10

12

85 6 743

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

f３ＫＯf

N３ＫＯ


UrSqu


14 Geofluids

minus0004

minus0003

minus0002

minus0001

0000

0001

0002

0003

0004

YD

minus15

minus10

minus05

00

05

10

15

CL

158154 156 160150 152

Time (s)


minus006

minus004

minus002

000

002

004

006

YD

158154 156 160150 152

Time (s)

minus15

minus10

minus05

00

05

10

15

CL



minus0004

minus0002

0000

0002

0004

YD

152 154 160158156150

Time (s)

minus10

minus05

00

05

10

CL


minus015

minus010

minus005

000

005

010

015YD

152 154 156 158 160150

Time (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

CL



minus0002

0000

0002

YD

151 158157156 159150 154153152 160155

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus0004

0004


minus04

minus03

minus02

minus01

00

01

02

03

04

YD

151 152 159154 158156 157 160150 153 155

Time (s)

minus25

minus20

minus15

minus10

minus05

00

05

10

15C

L



Geofluids 15

minus0006

minus0004

minus0002

0000

0002

0004

0006YD

151 152 153 154 155 156 157 158 159 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL


151 155153 157 159156152 158154 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus02

minus01

00

01

02

YD



V1

V1

V2

V2

S1


V3

V4

S2

V3

V4



tT３ＫＯ0 = 0 tT３ＫＯ0 = 0249

tT３ＫＯ0 = 0502 tT３ＫＯ0 = 075



tT３ＫＯ15 = 0 tT３ＫＯ15 = 025

tT３ＫＯ15 = 0497 tT３ＫＯ15 = 0752



16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions







effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)


00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (






Acknowledgments


Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

Geofluids 11

minus0006

minus0004

minus0002

0000

0002

0004

0006D

Squ

YSq

u

1815 21 2724 3330Time (s)


0

100

200

300

400

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus03

minus02

minus01

00

01

02

03

DSq

uY

Squ

18 20 2216Time (s)


0

5000

10000

15000

20000

25000

30000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)







12 Geofluids

minus02

minus01

00

01

02

DSq

uY

Squ

5 10 15 20 25 30 35 400Time (s)


0

5000

10000

15000

20000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus02

00

02

04

DSq

uY

Squ

3 6 9 12 150Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

2000

4000

6000

8000

10000

12000

14000

16000

Mag

nitu

de






Geofluids 13

minus02

minus01

00

01

02

DSq

uY

Squ

18 21 24 27 30 3315Time (s)


0

500

1000

1500

2000

2500

3000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000

005

010

015

DSq

uY

Squ

18 21 24 27 3015Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

4700

9400

14100

18800M

agni

tude



06

08

10

12

85 6 743

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

f３ＫＯf

N３ＫＯ


UrSqu


14 Geofluids

minus0004

minus0003

minus0002

minus0001

0000

0001

0002

0003

0004

YD

minus15

minus10

minus05

00

05

10

15

CL

158154 156 160150 152

Time (s)


minus006

minus004

minus002

000

002

004

006

YD

158154 156 160150 152

Time (s)

minus15

minus10

minus05

00

05

10

15

CL



minus0004

minus0002

0000

0002

0004

YD

152 154 160158156150

Time (s)

minus10

minus05

00

05

10

CL


minus015

minus010

minus005

000

005

010

015YD

152 154 156 158 160150

Time (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

CL



minus0002

0000

0002

YD

151 158157156 159150 154153152 160155

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus0004

0004


minus04

minus03

minus02

minus01

00

01

02

03

04

YD

151 152 159154 158156 157 160150 153 155

Time (s)

minus25

minus20

minus15

minus10

minus05

00

05

10

15C

L



Geofluids 15

minus0006

minus0004

minus0002

0000

0002

0004

0006YD

151 152 153 154 155 156 157 158 159 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL


151 155153 157 159156152 158154 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus02

minus01

00

01

02

YD



V1

V1

V2

V2

S1


V3

V4

S2

V3

V4



tT３ＫＯ0 = 0 tT３ＫＯ0 = 0249

tT３ＫＯ0 = 0502 tT３ＫＯ0 = 075



tT３ＫＯ15 = 0 tT３ＫＯ15 = 025

tT３ＫＯ15 = 0497 tT３ＫＯ15 = 0752



16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions







effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)


00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (






Acknowledgments


Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

12 Geofluids

minus02

minus01

00

01

02

DSq

uY

Squ

5 10 15 20 25 30 35 400Time (s)


0

5000

10000

15000

20000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus04

minus02

00

02

04

DSq

uY

Squ

3 6 9 12 150Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

2000

4000

6000

8000

10000

12000

14000

16000

Mag

nitu

de






Geofluids 13

minus02

minus01

00

01

02

DSq

uY

Squ

18 21 24 27 30 3315Time (s)


0

500

1000

1500

2000

2500

3000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000

005

010

015

DSq

uY

Squ

18 21 24 27 3015Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

4700

9400

14100

18800M

agni

tude



06

08

10

12

85 6 743

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

f３ＫＯf

N３ＫＯ


UrSqu


14 Geofluids

minus0004

minus0003

minus0002

minus0001

0000

0001

0002

0003

0004

YD

minus15

minus10

minus05

00

05

10

15

CL

158154 156 160150 152

Time (s)


minus006

minus004

minus002

000

002

004

006

YD

158154 156 160150 152

Time (s)

minus15

minus10

minus05

00

05

10

15

CL



minus0004

minus0002

0000

0002

0004

YD

152 154 160158156150

Time (s)

minus10

minus05

00

05

10

CL


minus015

minus010

minus005

000

005

010

015YD

152 154 156 158 160150

Time (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

CL



minus0002

0000

0002

YD

151 158157156 159150 154153152 160155

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus0004

0004


minus04

minus03

minus02

minus01

00

01

02

03

04

YD

151 152 159154 158156 157 160150 153 155

Time (s)

minus25

minus20

minus15

minus10

minus05

00

05

10

15C

L



Geofluids 15

minus0006

minus0004

minus0002

0000

0002

0004

0006YD

151 152 153 154 155 156 157 158 159 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL


151 155153 157 159156152 158154 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus02

minus01

00

01

02

YD



V1

V1

V2

V2

S1


V3

V4

S2

V3

V4



tT３ＫＯ0 = 0 tT３ＫＯ0 = 0249

tT３ＫＯ0 = 0502 tT３ＫＯ0 = 075



tT３ＫＯ15 = 0 tT３ＫＯ15 = 025

tT３ＫＯ15 = 0497 tT３ＫＯ15 = 0752



16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions







effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)


00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (






Acknowledgments


Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

Geofluids 13

minus02

minus01

00

01

02

DSq

uY

Squ

18 21 24 27 30 3315Time (s)


0

500

1000

1500

2000

2500

3000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


minus015

minus010

minus005

000

005

010

015

DSq

uY

Squ

18 21 24 27 3015Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

4700

9400

14100

18800M

agni

tude



06

08

10

12

85 6 743

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

f３ＫＯf

N３ＫＯ


UrSqu


14 Geofluids

minus0004

minus0003

minus0002

minus0001

0000

0001

0002

0003

0004

YD

minus15

minus10

minus05

00

05

10

15

CL

158154 156 160150 152

Time (s)


minus006

minus004

minus002

000

002

004

006

YD

158154 156 160150 152

Time (s)

minus15

minus10

minus05

00

05

10

15

CL



minus0004

minus0002

0000

0002

0004

YD

152 154 160158156150

Time (s)

minus10

minus05

00

05

10

CL


minus015

minus010

minus005

000

005

010

015YD

152 154 156 158 160150

Time (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

CL



minus0002

0000

0002

YD

151 158157156 159150 154153152 160155

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus0004

0004


minus04

minus03

minus02

minus01

00

01

02

03

04

YD

151 152 159154 158156 157 160150 153 155

Time (s)

minus25

minus20

minus15

minus10

minus05

00

05

10

15C

L



Geofluids 15

minus0006

minus0004

minus0002

0000

0002

0004

0006YD

151 152 153 154 155 156 157 158 159 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL


151 155153 157 159156152 158154 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus02

minus01

00

01

02

YD



V1

V1

V2

V2

S1


V3

V4

S2

V3

V4



tT３ＫＯ0 = 0 tT３ＫＯ0 = 0249

tT３ＫＯ0 = 0502 tT３ＫＯ0 = 075



tT３ＫＯ15 = 0 tT３ＫＯ15 = 025

tT３ＫＯ15 = 0497 tT３ＫＯ15 = 0752



16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions







effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)


00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (






Acknowledgments


Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

14 Geofluids

minus0004

minus0003

minus0002

minus0001

0000

0001

0002

0003

0004

YD

minus15

minus10

minus05

00

05

10

15

CL

158154 156 160150 152

Time (s)


minus006

minus004

minus002

000

002

004

006

YD

158154 156 160150 152

Time (s)

minus15

minus10

minus05

00

05

10

15

CL



minus0004

minus0002

0000

0002

0004

YD

152 154 160158156150

Time (s)

minus10

minus05

00

05

10

CL


minus015

minus010

minus005

000

005

010

015YD

152 154 156 158 160150

Time (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

CL



minus0002

0000

0002

YD

151 158157156 159150 154153152 160155

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus0004

0004


minus04

minus03

minus02

minus01

00

01

02

03

04

YD

151 152 159154 158156 157 160150 153 155

Time (s)

minus25

minus20

minus15

minus10

minus05

00

05

10

15C

L



Geofluids 15

minus0006

minus0004

minus0002

0000

0002

0004

0006YD

151 152 153 154 155 156 157 158 159 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL


151 155153 157 159156152 158154 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus02

minus01

00

01

02

YD



V1

V1

V2

V2

S1


V3

V4

S2

V3

V4



tT３ＫＯ0 = 0 tT３ＫＯ0 = 0249

tT３ＫＯ0 = 0502 tT３ＫＯ0 = 075



tT３ＫＯ15 = 0 tT３ＫＯ15 = 025

tT３ＫＯ15 = 0497 tT３ＫＯ15 = 0752



16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions







effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)


00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (






Acknowledgments


Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

Geofluids 15

minus0006

minus0004

minus0002

0000

0002

0004

0006YD

151 152 153 154 155 156 157 158 159 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL


151 155153 157 159156152 158154 160150

Time (s)

minus15

minus10

minus05

00

05

10

15

CL

minus02

minus01

00

01

02

YD



V1

V1

V2

V2

S1


V3

V4

S2

V3

V4



tT３ＫＯ0 = 0 tT３ＫＯ0 = 0249

tT３ＫＯ0 = 0502 tT３ＫＯ0 = 075



tT３ＫＯ15 = 0 tT３ＫＯ15 = 025

tT３ＫＯ15 = 0497 tT３ＫＯ15 = 0752



16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions







effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)


00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (






Acknowledgments


Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

16 Geofluids

tT３ＫＯ30 = 0

tT３ＫＯ30 = 0499

tT３ＫＯ30 = 0248

tT３ＫＯ30 = 0749


tT３ＫＯ45 = 0

tT３ＫＯ45 = 0499

tT３ＫＯ45 = 0252

tT３ＫＯ45 = 0751


6 Conclusions







effective area

(50ndash55)

(42ndash54)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV２－

３(V

)


00

10 times 10minus5

20 times 10minus5

30 times 10minus5

40 times 10minus5

50 times 10minus5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧＲ

)M

ax v

alue

of p

ower

gen

erat

ed (






Acknowledgments


Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

Geofluids 17

References
































18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in

18 Geofluids











Mining


Journal of









Journal of




Advances in











Geology Advances in








Mining


Journal of









Journal of




Advances in











Geology Advances in

Documents

Study on Fluid-Induced Vibration Power Harvesting of