and structures piezoelectric wind energy harvester for low...

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Journal of Intelligent Material Systemsand Structures22(18) 2215–2228! The Author(s) 2011Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1045389X11428366jim.sagepub.com

Piezoelectric wind energy harvesterfor low-power sensors

Jayant Sirohi and Rohan Mahadik

AbstractThere has been increasing interest in wireless sensor networks for a variety of outdoor applications including structuralhealth monitoring and environmental monitoring. Replacement of batteries that power the nodes in these networks ismaintenance intensive. A wind energy–harvesting device is proposed as an alternate power source for these wirelesssensor nodes. The device is based on the galloping of a bar with triangular cross section attached to a cantilever beam.Piezoelectric sheets bonded to the beam convert the mechanical energy into electrical energy. A prototype device ofsize approximately 160 3 250 mm was fabricated and tested over a range of operating conditions in a wind tunnel, andthe power dissipated across a load resistance was measured. A maximum power output of 53 mW was measured at awind velocity of 11.6 mph. An analytical model incorporating the coupled electromechanical behavior of the piezoelec-tric sheets and quasi-steady aerodynamics was developed. The model showed good correlation with measurements, andit was concluded that a refined aerodynamic model may need to include apparent mass effects for more accurate predic-tions. The galloping piezoelectric energy-harvesting device has been shown to be a viable option for powering wirelesssensor nodes in outdoor applications.

Keywordswind energy, energy harvesting, piezoelectric, galloping

Introduction

The availability of low-power microprocessors and sen-sors, in conjunction with data loggers and wireless com-munication, is enabling a wide range of distributedsensing applications. An example of such an applicationis a network of sensor nodes distributed over a largecivil structure, such as a bridge (Kim et al., 2007; Wanget al., 2006), where each node senses local parameterssuch as vibration amplitude or strain. These data canbe transmitted wirelessly to a base station or storedlocally for future interrogation. In this way, the state orhealth of the structure can be monitored. Wireless sen-sor networks are also used in a number of environmen-tal monitoring applications (Badrinath et al., 2000;Estrin et al., 2000; Evans and Bergman, 2007; Wang etal., 2006).

Yick et al. (2008) discussed several commerciallyavailable wireless sensors, their vendors, and theirapplications. The energy requirement of each node istypically small and can be met by a battery pack. Forexample, Mainwaring et al. (2002) described a sensornetwork in an ecological reserve to monitor seabirdnesting, in which each sensor node required 6.9 mAhper day.

Replacement of depleted batteries for large sensornetworks can be expensive, time consuming, and envir-onmentally unfriendly. To overcome these issues, therehas been increasing interest in the use of energy-harvesting methods to power the sensors on-site. Thisapproach eliminates the need for batteries, and theassociated requirement of periodic replacement and dis-posal. This is especially important in outdoor locationswith inaccessible terrain, or geological parks wherehuman interaction is minimum and environmentalimpact is key. There have been numerous studies onharvesting the energy in ambient structural vibrationsusing piezoelectric materials. Because a majority ofwireless sensors are located outdoors, powering themby means of wind could also be a practical alternative.Aeroelastic instabilities such as flutter and gallopinghave been explored for harvesting energy from wind.

Department of Aerospace Engineering and Engineering Mechanics, TheUniversity of Texas at Austin, Austin, TX, USA

Corresponding author:Jayant Sirohi, Department of Aerospace Engineering and EngineeringMechanics, The University of Texas at Austin, Austin, TX 78712, USAEmail: [email protected]

Energy Harvesting Using Piezoelectric Materials

The electromechanical coupling exhibited by piezoelec-tric materials can be harnessed to extract electricalenergy from mechanical vibrations. As a result, piezo-electric materials have found wide application as low-power generators. In a majority of these applications,the piezoelectric material extracts energy from ambientstructural vibrations by operating as a base-excitedoscillator. Sodano et al. (2004) provided an overview ofseveral studies related to piezoelectric energy harvest-ing, including devices based on impact; wearableenergy-harvesting devices based on motion of thehuman body; and devices designed to power wirelesssensors. They also discussed methods to accumulate theharvested energy, using rechargeable batteries, capaci-tors, or flyback converters. While most of the energy-harvesting devices are based on cantilever beams, othergeometries such as annular piezoelectric unimorphs/bimorphs have also been explored (Kauffman andLesieutre, 2009).

duToit et al. (2005) investigated vibration-basedpiezoelectric energy harvesters to power MEMS-scaleautonomous sensors. They compared the power densityof electrostatic, electromechanical, and piezoelectricvibration-based energy harvesters and concluded thatthe piezoelectric devices have the highest power densitybased on volume.

Vibration-based piezoelectric energy harvesters arelimited to relatively low-power outputs, on the order of1–1000 mW (see duToit et al., 2005) due to the inher-ently low levels of strain energy in structural vibrations.Therefore, optimizing the power conditioning and stor-age electronics is an important part of the overall device(see Ottman et al., 2003).

It is interesting to note that vibrational energy har-vesting using piezoelectric materials is closely related topiezoelectric shunt damping because both conceptsextract energy from the structure, resulting in an effec-tive negative damping. However, in the case of energyharvesting, the goal is to accumulate the energy whilein the case of shunt damping, the goal is to dissipate asmuch of the energy as possible.

Energy Harvesting from Aeroelastic Instabilities

Some piezoelectric energy-harvesting devices have beendeveloped to harness energy from structural vibrationsinduced by wind (Tan and Panda, 2007; Wang and Ko,2010). Robbins et al. (2006) investigated the use of flex-ible, flag-like, piezoelectric sheets to generate powerwhile flapping in an incident wind. The energy that canbe harvested using these approaches is comparable tothat of a vibration-based device. By exploiting struc-tures with aeroelastic instabilities, it is possible toextract significantly higher amounts of energy from thewind. Bryant and Garcia (2011) developed a device to

harvest energy from flutter, using a piezoelectricbimorph with a flap at its tip. Linear and nonlinearmodels were developed to predict the performance ofthe device. The device generated an output power onthe order of 2 mW.

Galloping is an aeroelastic instability involving low-frequency, large-amplitude oscillations of the structurenormal to the direction of incident wind. Typically, itoccurs in lightly damped structures with asymmetriccross sections, such as ice-covered transmission lines.There have been several studies on the effect of variousparameters influencing galloping behavior of prismaticstructures with different cross sections (Blevins, 2001),such as rectangular (Kazakevich and Vasilenko, 1996),D-section (Laneville et al., 1977; Ratkowski, 1961), tri-angular (Alonso et al., 2005, 2007; Alonso andMeseguer, 2006), and elliptic (Alonso et al., 2010).Barrero-Gil et al. (2010) theoretically investigated thefeasibility of energy harvesting from structures under-going galloping. They represented the sectional aerody-namic characteristics using a cubic polynomialand obtained an expression for the harnessable energy.Specific methods for energy extraction were notdiscussed. Nondimensional parameters defining theachievable power density and efficiency of energy con-version were derived. Sirohi and Mahadik (2011) inves-tigated wind energy harvesting using a beam withpiezoelectric sheets attached to a galloping tip bodywith D-shaped cross section. The variation of thepower generated by the piezoelectrics was measured asa function of wind speed. The device produced a maxi-mum power on the order of 0.5 mW.

This article describes a wind energy–harvestingdevice based on a galloping triangular section attachedto cantilever beams with surface-bonded piezoelectricsheets. A prototype is tested and the results correlatedwith an analytical model. Such a device has the primaryadvantages of simplicity and robustness and could becollocated with the outdoor wireless sensors to powerthem using renewable wind energy.

Galloping Piezoelectric Energy Harvester

The galloping piezoelectric energy-harvesting deviceinvestigated in this article consists of a prismatic rigidbody attached to the tip of two aluminum cantileverbeams that act like a flexible support. Piezoelectricsheets are bonded near the root of the beams, at thelocation of maximum strain energy. Galloping of theprismatic body in an incident wind results in large-amplitude oscillatory bending of the beams that is con-verted into electrical energy by the piezoelectric sheets.While the tip body can have any cross section that isprone to galloping, the device described in this articlehas a tip body with an equilateral triangle cross section(Figure 1). Wind incident on the flat face of the

2216 Journal of Intelligent Material Systems and Structures 22(18)

triangular section results in galloping motion and bend-ing of the cantilever beams.

Physical Principles

Den Hartog (1956) explained the phenomenon of gal-loping for the first time in 1934 and introduced a criter-ion for galloping stability of a structure. He developeda criterion for galloping to occur on a section havingspecified lift and drag coefficients, on a flexible supportwith mechanical damping. The Den Hartog stabilitycriterion states that a section on a flexible support withnegligible mechanical damping is susceptible to gallop-ing when

H a! " = dCl

da+Cd

! ".0 !1"

where Cl and Cd are the sectional lift and drag coeffi-cients, respectively, and a is the sectional angle ofattack. The quantity H(a) is referred to as the DenHartog factor. Note that galloping occurs at highangles of attack where the aerodynamic coefficients arehighly nonlinear. Therefore, the criterion is evaluatedby considering a linearized slope of Cl versus a at theequilibrium point about which oscillations occur. Theonset of galloping is characterized by a negative effec-tive damping of the system and corresponding expo-nential increase in the amplitude of motion with time.However, the system reaches a limit cycle oscillation ina short period of time, after which the amplitude of

oscillation remains constant. Because the frequency ofoscillation is relatively low, it has been observed thatquasi-static aerodynamics are sufficient to model thebehavior (Alonso et al., 2007; Barrero-Gil et al., 2010).

Consider a coordinate system with the x-axis alongthe span of the beam, y-axis along the beam width, andz-axis along the beam thickness. The bending displace-ment of the beam is given by w(x). The sectional angleof attack of the tip body is then given by

a=a0 # tan#1 _wtip

V‘

! "!2"

where a0 is the initial angle of attack (when the sectionis at rest) with respect to the axes along which the sec-tional aerodynamic coefficients were measured, V‘ isthe magnitude of the incident wind (Figure 2), and _wtip

is the velocity of the beam tip.At the steady-state condition, the extracted energy is

equal to the work done on the system by the incidentwind. Barrero-Gil et al. (2010) calculated a theoreticalvalue of around 7% for the efficiency of convertingwind energy into mechanical energy by galloping.Although this is low compared to the Betz limit forwind turbines, galloping energy harvesters can be analternative in cases where robust, low-maintenance,low-power energy generation devices are required. Therange of angles of attack over which galloping occursindicates that the device is self-limiting in terms ofamplitude as well as wind speed. Theoretically, themaximum amplitude of oscillation corresponds to asectional angle of attack at which the Den Hartog cri-terion is no longer satisfied. Therefore, no additionalprecautions need be taken to protect the device in caseof excessive wind speed.

Parameters of the Prototype Device

A prototype device was constructed for measurementof the power output and correlation with analysis.In addition to the beam geometry, the sectional

Figure 1. Galloping energy harvester with tip body havingequilateral triangle cross section.

Figure 2. Incident velocity and coordinate system of section.

Sirohi J and Mahadik R 2217

aerodynamic coefficients of the tip body are key to theperformance of the device. Two identical aluminumbeams were clamped to a support at one end and to aprismatic tip body at their other end. Two piezoelectricsheets (PSI-5H4E from Piezo Systems, Inc.) of length72.4 mm, width 36.2 mm, and thickness 0.267 mm werebonded to the top and bottom surfaces of each beamwith a high shear strength epoxy adhesive. After curing,the bond layer was measured to be approximately0.025 mm thick. The piezoelectric sheets were con-nected in parallel with opposite polarity, because thetop layer undergoes tension when the bottom layer isundergoing compression. In this way, the charges devel-oped by the piezoelectric sheets are added together, andthe effective capacitance is the sum of the capacitancesof the individual sheets. The properties of the beam, tipbody, and piezoelectric sheets are listed in Table 1.

An equilateral triangle section was chosen for the gal-loping device because it exhibits a favorable combinationof a large range of operating angles of attack as well ashigh Den Hartog factor (see Alonso andMeseguer, 2006).

The sectional aerodynamic coefficients of an equilat-eral triangle section were measured by Alonso andMeseguer (2006) and are shown in Figure 3. The angleof attack for these data is with reference to a line ofsymmetry of the triangular section. From the figure, itis seen when the wind is incident normal to a face ofthe triangle (angle of attack of 60" with respect to a lineof symmetry), the slope of Cl is negative, and Cd is rela-tively constant. In this condition, the Den Hartog con-dition is satisfied and the section is prone to galloping.It is also seen that the variation of aerodynamic coeffi-cients with angle of attack is symmetric with respect toa line of symmetry of the triangle.

Analytical Model

Several analytical models have been proposed to quan-tify the electrical output from a piezoelectric energyharvester. Sodano et al. (2004) used Hamilton’s princi-ple to develop a coupled electromechanical model for apiezoelectric bimorph energy-harvesting device. Umedaet al. (1996) proposed an electrical equivalent circuit to

model the electrical energy generated by a metal ballfalling on a plate with a bonded piezoelectric sheet.Roundy et al. (2004) discussed the operation and analy-tical modeling of a base-excited energy harvester con-sisting of a piezoelectric bimorph with a lumped massattached to its tip. Ajitsaria et al. (2007) developedthree different mathematical models for the voltagegenerated by a bimorph piezoelectric cantilever beam.Erturk and Inman (2009) modeled their device by cou-pling an Euler–Bernoulli beam representation with thepiezoelectric constitutive laws.

In this article, an analytical model of the gallopingenergy-harvesting device is developed using a varia-tional approach, in conjunction with assumed displace-ments and quasi-steady aerodynamic forcing. The goalof the model is to predict the voltage generated by thepiezoelectric sheets as a function of time, for a givenincident wind speed, beam geometry, and load resis-tance. This study focuses on the conversion of windenergy into electrical energy and does not consider stor-age of the generated electrical energy. Accordingly, theelectrical load on the system is represented by a loadresistance connected across the electrodes of the piezo-electric sheets, which dissipates energy in the form ofOhmic heating and acts as an additional positive damp-ing on the system. Therefore, the amplitude of oscilla-tion of the beam increases till a steady-state limit cycle isreached, wherein the energy extracted from the incidentwind is exactly equal to the energy dissipated across theload resistance and by any structural damping.

The analytical model consists of two parts: the struc-tural model and the aerodynamic model. Energy-harvesting devices of similar geometry and having

Table 1. Parameters of the prototype galloping wind energyharvester.

Parameter Measurement (mm)

Beam length 161Beam width 38Beam thickness 0.635Tip body length 251Tip body dimension 40 (side)Piezo sheet length 72.4Piezo sheet width 36.2Piezo sheet thickness 0.267

Figure 3. Aerodynamic coefficients of an equilateral trianglesection (Alonso and Meseguer, 2006).


different tip bodies can be modeled by simply changingthe sectional aerodynamic coefficients.

Structural Model

A schematic of a single beam with the triangular tipbody, indicating the coordinate directions and relevantdimensions, is shown in Figure 4. The device is mod-eled as two Euler–Bernoulli beams attached to a tipmass. To minimize blockage of the incident wind, thedevice must be clamped to a support with a low frontalarea. Consequently, the support will have a lower stiff-ness than an ideal clamped boundary condition. Theflexibility of the support is modeled by incorporating atorsional spring of stiffness ku (N/m) at the root of thebeam.

The analytical model of the electromechanical cou-pling of the piezoelectric sheets is developed using anenergy-based variational formulation. The basic proce-dure is to formulate a variational indicator incorporatingthe kinetic energy, potential energy, and nonconservativevirtual work on the system. The potential energyincludes contributions from the strain energy as well asstored electrical energy. Similarly, the nonconservativevirtual work includes mechanical and electrical terms.The variational indicator can be set up in two waysbased on the choice of independent variables, asdescribed by Crandall et al. (1982).

In one approach, the variational indicator (V :I :) iswritten as

V :I :=

#t2

t1

$d T # V #We! "+X

i

fidwi +X

j

Vjdqj%dt

=

#t2

t1

$d T #U! "+X

i

fidwi +X

j

Vjdqj%dt

!3"

where T is the kinetic energy of the structure, V is thestrain energy, and We is the electrical energy. The sum-mations represent the virtual work done by all noncon-servative mechanical and electrical elements in the

system. In the present case, fi are the transverse forcesapplied to the beam, wi are the transverse displace-ments, Vj is the voltage drop across the nonconserva-tive electrical elements (in this case, the load resistance),and qj is the electric charge. The strain energy and elec-trical energy can be combined into an internal energy Ugiven by

U !,D! "=V +We

=1

2

#

Vs

sT! dVs +1

2

#

Vs

ETD dVs!4"

where ! is the strain vector, s is the stress vector, D isthe electric displacement vector, E is the electric fieldvector, and Vs is the volume of the structure. Note thatthe internal energy must be expressed as a function ofindependent variables corresponding to displacementand charge, which are in this case, ! and D (see Mason,1950). Several researchers have adapted this approachto model the electromechanical coupling in structureswith piezoelectric material (duToit et al., 2005; Elvinet al., 2001; Hagood et al., 1990; Sodano et al., 2004).

Dietl and Garcia (2010) used an alternate approach,based on flux linkage, to optimize the shape of a piezo-electric beam for energy harvesting. In this approach,the variational indicator is written as

V :I :=

#t2

t1

$d T # V +W &e

$ %+X

i

fidwi +X

j

ijdlj%dt

=

#t2

t1

$d T # H2! "+X

i

fidwi +X

j

ijdlj%dt

!5"

where W &e is the electrical co-energy and ij are the cur-

rents flowing through the dissipative electrical elementsin the system. The lj are the flux linkages, which arerelated to voltages by

V= _l !6"

The strain energy and electrical co-energy can becombined into an electrical enthalpy H2 given by

H2(!,E) =V #W &e

=1

2

#

Vs

sT! dVs #1

2

#

Vs

DTE dVs!7"

Note that the electrical enthalpy must be expressedas a function of independent variables correspondingto displacement and electric field, which are in thiscase, ! and E (see Mason, 1950). It can be shown thatthe approaches based on the two variational indicatorsare equivalent because the internal energy and electricenthalpy are related to each other by a Legendretransformation.

L1

L2Lb

Piezoelectricsheets

x

z

tbtp

k!

Figure 4. Schematic of beam with piezoelectric sheets and tipbody.


The formulation based on internal energy (Equation(3)) is used for the present analysis. The constitutiverelations of the piezoelectric material are given as(IEEE, 1987)

!

D

& '=

sE dT

d es

( )sE

& '!8"

where d is the matrix of piezoelectric coefficients, s isthe compliance matrix, and e is the dielectric permittiv-ity matrix. The superscripts E and s refer to quantitiesmeasured at constant electric field and constant stress,respectively. The piezoelectric constitutive relations canbe rearranged in terms of the strain and electric displa-cement as

sE

& '=

cD #hT

#h be

( )!D

& '!9"

In the case of the present device, the piezoelectricsheets are attached so that their 1-axis is along thelength of the beam (x-direction) and the 3-axis isalong the thickness of the beam (z-direction). Strainsalong the y-direction are ignored, and a one-dimensional representation is used to model thedevice. Reducing Equation (9) to one dimension andsubstituting the relevant piezoelectric constants fromEquation (8) yield

s11

E3

& '=

YD11 # 1

d931# 1

d9311ee33

" #e11D3

& '!10"

where

YD11 =

YE11

1# k231=

1

sE11 1# k231$ % !11"

d931 =d31 1# k231

$ %

k231!12"

ee33 = es 1# k231$ %

!13"

The superscripts D and e refer to quantities measuredat constant electric displacement and constant strain,respectively. The quantity Y11 is the Young’s modulusof the piezoelectric material. In these equations, theelectromechanical coupling factor of the piezoelectricsheets is defined as

k231 =d231Y

E11

es33!14"

It is convenient to model the coupled behavior ofthe piezoelectric sheets in this way because it is rela-tively simple to measure the constants YE

11, d31, and es33.Substituting these quantities into Equation (5) yieldsthe internal energy of the device as

U =1

2

#

Vs

s11e11 dVs +1

2

#

Vs

D3E3 dVs

=1

2

#

Vs

YD11e

211 +

D23

ee33

! "dVs #

#

Vs

D3e11d931

dVs

!15"

The integration is performed over the volume of theentire structure, taking care to set the appropriate mate-rial constants over the piezoelectric elements and thealuminum beam. Applying the Euler–Bernoulli assump-tion to the aluminum beams results in the longitudinalstrain given by

e11 = # zw99 !16"

A superposition of assumed shape functions f(x)and generalized displacement coordinates r(t) is used torepresent the transverse deflection as

w x, t! "=XN

i= 1

f x! "r t! "=fr !17"

which gives the longitudinal strain as

e11 = # zf99r !18"

Note that the potential energy stored in the torsionalspring (of stiffness ku in Newton meters per radian) atthe root of the beam, given by Vspring, must be added tothe internal energy of the structure.

Vspring =1

2ku w9 0! "$ %2 !19"

The angular deflection at the root of the beam isgiven by

w9 0! " =f9 0! "r !20"

The electric displacement can be represented as asummation of assumed functions c(z) and generalizedcharge coordinates q(t) as

D z, t! "=XM

j= 1

c z! "q t! "=cq !21"

Note that if the electric field across the piezoelectricsheets is assumed to be a constant:

c=1

Ap!22"

where Ap is the area of the electrodes on the piezoelec-tric sheets. In this case, the electric displacement is givenby

D=q

Ap!23"


where q is the physical charge generated by the piezo-electric sheets. The assumption of constant electric fieldacross the piezoelectric sheets will be retained for theremainder of this analysis. Substituting the strain andelectric displacement into Equation (15), and includingthe potential energy of the root spring, the internalenergy can be written as

U =1

2rTKr+

1

2

q2

Cp+ rT"q !24"

where the stiffness matrix K and the coupling matrix "

are given by

K=

#

Vp

YD11z

2f99Tf99 dVp

+

#

Vb

Ybz2f99Tf99 dVb +f9 0! "T kuf9 0! "

!25"

"=

#

Vp

zf99T

d931ApdVp !26"

The subscripts b and p denote quantities corres-ponding to the beam and the piezoelectric sheets,respectively.

Note that the capacitance of the piezoelectric sheets(at constant strain) is given by

Cp =ee33Ap

tp!27"

The kinetic energy of the structure is given by

T =1

2

#

Vs

r _w2 dVs +1

2Mtip _w Lb! "! "2

=1

2_rTM_r !28"

where the mass matrix is

M =

#

Vb

rbfTf dVb +

#

Vp

rpfTf dVp +f Lb! "TMtipf Lb! "

!29"

In the case of the energy-harvesting device under con-sideration, the nonconservative mechanical virtual workarises only due to the aerodynamic force Ftip acting onthe tip body. The nonconservative electrical virtual workis the energy dissipated by the load resistance RL.

Substituting the internal energy (Equation (24)),kinetic energy ((Equation (28)), and nonconservativevirtual works into Equation (3) and setting the varia-tional indicator to zero,

V :I :=

#t2

t1

d T # U! "+Ftipdw Lb! " +Vdq* +dt = 0 !30"

yields the equations of motion of energy-harvestingdevice as

M!r+Kr+"q=Ftipf Lb! " !31"

"T r+1

Cpq# V= 0 !32"

In the present case, the voltage drop across the loadresistance is given by

V= # RL _q !33"

The mechanical damping of the structure is incorpo-rated into the model in terms of a proportional damp-ing matrix (see Sodano et al., 2004) C given by

C=aM+bK !34"

where the constants a and b are determined from

zk =a

2vk+bvi

2(i= 1, 2 . . .N ) !35"

In the above equation, vk is the natural frequency ofthe kth mode, zk is the modal damping, and N is thenumber of modes (equal to the dimension of the massand stiffness matrices). These quantities are measuredexperimentally from the impulse response of the beamwith an appropriate electrical boundary condition forthe piezoelectric sheets. An additional damping is intro-duced due to the energy dissipation in the internal resis-tance of the piezoelectric sheets, Ri. This resistance is afunction of the dissipation factor (expressed as tan d)of the piezoelectric material. For small values of tan d,the internal resistance can be written as (Sirohi andChopra, 2000)

Ri =tan d

vCp!36"

where v is the frequency of voltage across the electro-des of the piezoelectric sheet. At large values of electricfield, the dissipation factor and other piezoelectric con-stants become nonlinear functions of the electric field.However, this effect is neglected in the present analysis.The parameters of the piezoelectric sheets used in theanalysis are listed in Table 2.

The equations of motion can be written in the statespace form by defining a state vector containing the

Table 2. Parameters of piezoelectric material

Piezoelectric property Symbol Value

Strain coefficient (pC/N) d31 2320Young’s modulus (GPa) YE11 62Dielectric constant (nF/m) es33 33.65Density (kg/m3) rp 7800


generalized displacement (consisting of N assumedmodes), generalized velocity, and charge:

x=r_rq

8<

:

9=

; !37"

The equations of motion (Equations (32) and (33))can then be written as

_x=

0(N3N ) I(N3N ) 0(N31)

#M#1K #M#1C #M#1"

# 1RL"T 0(13N ) # 1

RLCp

2

4

3

5x

+0(N31)

M#1f Lb! "T0

8<

:

9=

;Ftip !38"

The present analysis assumes a two-term approxima-tion for the transverse deflection as

w x, t! "=f1r1 +f2r2 =x

Lb

! "r1 +

x

Lb

! "3

r2 !39"

The first assumed mode represents the deformationof the support (root torsional spring), and the secondmode represents the bending of the beam. The assump-tion of a single bending deflection shape is consistentwith visual observations of the beam deformation dur-ing galloping. In addition, the system is expected to gal-lop at a frequency close to its first mode, andconsequently, a shape function proportional to its staticdeflection is chosen to approximate this mode shape.

Substituting the assumed deflection and electric dis-placement into Equations (25), (26), and (30), we getexpressions for the effective mass and stiffness matrices,and the electromechanical coupling matrix.

M11 =rbbbtbLb

3+2rpbptp L32 # L31

$ %

3L2b+Mtip !40"

M12 =M21 =rbbbtbLb

5+2rpbptp L52 # L51

$ %

5L4b+Mtip !41"

M22 =rbbbtbLb

7+2rpbptp L72 # L71

$ %

7L6b+Mtip !42"

K11 =ku

L2b!43"

K12 =K21 = 0 !44"

K22 =Ybbbt

3b

L3b+8YD

11bp L32 # L31$ %

L6b

tb

2+ tp

, -3

# tb

2

, -3( )

!45"

Y1 = 0 !46"

Y2 =3d31Y

E11bp

L3bCptptp tb + tp$ %

L22 # L21$ %

!47"

Aerodynamic Model

In the present galloping device, the force acting on thetip body is calculated assuming quasi-static, two-dimensional aerodynamics. The vertical sectional forceper unit length (Fz) is multiplied by the length of the tipbody (Ltip) to give the total tip force (Ftip). Componentsof the sectional lift L and sectional drag D per unitlength contribute to the force Fz.

Ftip =FzLtip = L cos a+D sin a! "Ltip !48"

The quasi-static angle of attack at each section of thetip body is given by

a= tan#1 _w Lb! "V‘

! "+w9 Lb! " !49"

where the first term is due to the relative wind velocityand the second term is due to the structural deforma-tion. The sectional lift and drag per unit length aregiven by

L=1

2r V 2

‘ + _w2 Lb! "$ %

btipCl !50"

D=1

2r V 2

‘ + _w2 Lb! "$ %

btipCd !51"

where Cl and Cd are the sectional lift and drag coeffi-cients, respectively; btip is the width of the tip body; andr is the density of air. The sectional aerodynamic coef-ficients depend on the quasi-static angle of attack aswell as Reynolds number. The coefficients that wereused in the analysis are plotted in Figure 3. Note thatthese data were measured at a different Reynolds num-ber than the present experiment. This could be a sourceof discrepancy while correlating predictions based onthese coefficients with measurements. Recall that theangle of attack for these data is indexed to a line ofsymmetry of the triangular section. Because the wind isincident on a face of the triangle in the present device,a constant offset of ao = 608 must be added to the angleof attack calculated from Equation (50) to obtain theangle of attack corresponding to Figure 3.

Solution Procedure

The governing equations of the system were solved bytime marching using MATLAB. A small displacementof the beam (r1 = 0:00100) was set as an initial condition.All initial velocities and initial charge were set to 0. Thiscorresponded to an initial angle of attack of approxi-mately 1". It was found that small variations in the initialangle of attack in this range did not affect the solution.

At each time step, the angle of attack calculatedfrom Equation (50) was used to perform a table lookupto find the aerodynamic coefficients. These were thenused to find the sectional lift and drag, from which the


force on the tip body was calculated. The block dia-gram shown in Figure 5 summarizes the solution proce-dure. The model predicts beam displacement w(x, t)and power generated for given values of incident windvelocity, load resistance, and device geometry. Tip bod-ies of any cross-sectional geometry can be evaluated byincorporating the appropriate sectional airfoil charac-teristics in the form of a table lookup.

Experimental Setup and Procedure

A prototype galloping device was constructed andtested to validate the analytical model. The parametersof the device are listed in Table 1. The tip body wasconstructed out of rigid foam and mounted on thebeams such that its angle of incidence could be manu-ally adjusted. The prototype was assembled in the testsection of a closed-circuit, open-jet wind tunnel. Theinstallation is shown in Figure 6, in which the triangu-lar section is aligned such that it does not gallop.

The power generated by the galloping device wasmeasured at several wind velocities. At a specific value

of wind velocity, a load resistance was connected acrossthe electrodes of the piezoelectric sheets. The tip bodywas manually rotated so that the wind was incident nor-mal to a flat face, and then released. Small perturbationsgenerated by vortex shedding and turbulence in the air-stream caused the device to start galloping. The voltagedeveloped across the load resistance as a function of timewas measured. After the voltage reached a steady state,the galloping was manually arrested and the procedurerepeated with a different load resistance. The power gen-erated was calculated over a range of incident windspeeds by varying the load resistance across the piezo-electric sheets and measuring the voltage across them ateach wind speed. A NI 9205 data acquisition system inconjunction with Labview was used to acquire the data.Because the generated voltage was, in most cases, greaterthan 10 V, a voltage divider circuit was used to scale itdown to the range of the data acquisition system. Theresistance of the voltage divider was chosen to be muchhigher than the output impedance of the piezoelectricsheets so that it did not contribute to the load resistance.In addition, the output of the voltage divider was buf-fered by a unity gain operational amplifier so that itwould not be affected by the input impedance of thedata acquisition system. The wind speed was measuredby a conventional pitot probe and manometer.

Results and Discussion

Prior to measuring the power output, the effectivedamping of the system was measured by the logarith-mic decrement technique, for a range of values of loadresistance. The value of load resistance at which theeffective damping doubles (compared to the open cir-cuit case) is approximately equal to the output impe-dance of the piezoelectric sheets, neglecting structuraldamping in the beam. Figure 7 shows the measured

Figure 5. Block diagram of solution procedure.

Open-jet wind tunnel

Pitotprobe

PiezoelectricsheetsTriangular

section

Wind

Figure 6. Prototype galloping device installed in the wind tunneltest section.

Figure 7. Variation of damping ratio with load resistance (in theabsence of wind).


damping ratios, from which the output impedance ofthe piezoelectric sheets was determined to be between25 and 40 kO. This is a convenient method to experi-mentally estimate the output impedance of the piezo-electric sheets, and therefore, the load resistance atwhich maximum power generation is expected.

Based on the damping ratio measurements, a rangeof load resistances from 5 to 300 kO was chosen for thepower measurements. The transient voltage generatedby the device was recorded with these load resistances atdifferent values of incident wind velocity. Figure 8shows the transient voltage measured at a wind velocityof 10 mph and a load resistance of 37 kO. The initialperturbations are seen to grow in amplitude till a steadystate is reached. Note that the mean voltage at steadystate is slightly negative. This can be attributed to errorsarising from manual setting of the initial angle of attack.If the initial angle of attack is not exactly 0", there willbe a bias aerodynamic force acting on the tip body,which will result in galloping about a slightly deformedmean position. Also note that the amplitude of thesteady-state response appears to contain an oscillatorycomponent, which may be due to turbulence in the inci-dent airstream. The maximum voltage generated isgreater than 30 V (corresponding to an electric field ofmore than 157 kV/m), which is relatively high consider-ing that the piezoelectric material constants (Table 2)are valid only in a linear region at small electric fields.

The output voltage predicted by the analysis at awind velocity of 10 mph and a load resistance of 37 kOis shown in Figure 9. The trend of the predicted voltageclosely approximates that of the measured voltage.Some discrepancy can be seen in the transient response;however, the steady-state behavior correlates very wellwith predictions. Note that the predicted voltage has amean of zero and does not show any oscillatory varia-tion of the steady-state amplitude.

It is interesting to explore the calculated angle ofattack of the tip body and the tip displacement at thisoperating condition. The angle of attack (referencedto a line of symmetry of the tip body), shown in

Figure 10, increases from around the mean of 60" till itreaches a steady state where it oscillates between a min-imum of 16.7" and a maximum of 103.4". A plot of theDen Hartog factor, H( =dCl=da+Cd), calculated fromthe measured sectional aerodynamic coefficients (seeFigure 3) is shown in Figure 11. The small excursionsaround a= 408 and a= 808 could be due to experimentalerrors. It is seen that the Den Hartog factor is negativeover a range of angles of attack approximately258\a\958, which is a smaller range than calculatedfor the present device (Figure 10). This indicates thatthe natural frequency of the present device is highenough that dynamic effects could be important. Inaddition, the analysis indicated that the peak-to-peakangle of attack variation at steady state increased withincreasing incident wind velocity (Figure 12). Note thatthe peak-to-peak angle of attack variation correspond-ing to a negative Den Hartog factor is 708.

The predicted tip displacement is shown in Figure13. The maximum amplitude of the tip displacement isapproximately 45 mm, which is considerable for a beamof length 167 mm. Durability of the device under suchlarge fatigue loading could be a matter of concern andmust be investigated in greater detail. In the case of theprototype device, several cracks developed across the

Time (s)

Out

put v

olta

ge (V

)

0 2 4 6 8 10 12 14 16 18 20-40

-30

-20

-10

0

10

20

30

40

Figure 8. Measured output voltage, VN = 10 mph, Rl = 37 kO.

Time (s)

Out

put v

olta

ge (V

)

0 2 4 6 8 10 12 14 16 18 20-40

-30

-20

-10

0

10

20

30

40

Figure 9. Predicted output voltage, VN = 10 mph, Rl = 37 kO.

Time (s)

Ang

le o

f atta

ck (°

)

0 2 4 6 8 10 12 14 16 18 2010

20

30

40

50

60

70

80

90

100

110

Figure 10. Predicted angle of attack of the tip body (referencedto a line of symmetry), VN = 10 mph, Rl = 37 kO.


width of the piezoelectric sheets after a few experimen-tal runs. Consequently, Young’s modulus of the piezo-electric material had to be reduced to approximately50% in the analytical model to account for the loss instiffness.

The measured output power (peak value) over therange of load resistances is shown in Figure 14. It isseen that the maximum output power is achieved at a

load resistance of approximately 37 kO, which is closeto the impedance of the piezoelectric sheets. Asexpected, the output power increases with increasingwind velocity. A maximum output power of 53 mWwas measured, which is sufficient to power most of thewireless sensor nodes currently in use. This maximumpower is significantly higher than that measured in aprevious study of a galloping device (Sirohi andMahadik, 2011) and is also much higher than achiev-able by means of vibrational energy harvesting.

The measured output power (peak value) over therange of wind velocities is shown in Figure 15. Thedevice starts to extract power at wind speeds higherthan approximately 8 mph. Therefore, this velocity canbe considered as a cut-in velocity for extraction ofpower. However, the power abruptly drops to almost 0for a wind velocity of 13.6 mph. This unexpected beha-vior is attributed to large-scale turbulence in the inci-dent wind stream.

The correlation between experimental measurementsand analytical predictions is shown in Figure 16 for anincident wind speed of 10 mph. The output powerincreases with increasing load resistance, reaches a max-imum, and then decreases. The analysis is able to

Angle of attack (°)

Den

Har

tog

Fact

or H

0 20 40 60 80 100 120 140 160 180-2

-1

0

1

2

3

4

5

Figure 11. Den Hartog factor calculated from sectional aerody-namic coefficients measured by Alonso and Meseguer (2006).

Incident wind velocity (mph)

Peak

-to-

peak

ang

le o

f atta

ck (°

)

8 9 10 11 12 1385

86

87

Figure 12. Predicted peak-to-peak angle of attack variation atsteady state, Rl = 37 kO.

Time (s)

Tip

disp

lace

men

t (m

m)

0 2 4 6 8 10 12 14 16 18 20-50

-40

-30

-20

-10

0

10

20

30

40

50

Figure 13. Predicted tip displacement, VN = 10 mph, Rl =37 kO.

0

10

20

30

40

50

0 50 100 150 200 250 300

11.6 mph

10 mph

8.2 mph

Load resistance (k")

Out

put p

ower

(mW

)

Figure 14. Variation of measured output power with loadresistance.

0

10

20

30

40

50

0 2 4 6 8 10 12

Wind velocity (mph)

Out

put p

ower

(mW

)

5 k"15.8 k"36.9 k"50 k"100 k"

14

Figure 15. Variation of measured output power with incidentwind velocity.


capture the trend very well, but overpredicts the magni-tude of the output power. This discrepancy can beattributed to several causes. First, the bond layerbetween the piezoelectric sheets and the beam is notmodeled in the present analysis. The finite stiffness ofthe bond layer can cause a significant decrease in theenergy transferred to the piezoelectric sheets. Second,the quasi-static aerodynamic analysis does not includeany apparent mass effect of the air. This can be seenfrom the measured frequency of oscillations of thebeam. In the absence of incident wind, the natural fre-quency of the beam was measured to be 10.43 Hz at aload resistance of 33.5 kO. The natural frequency variesby approximately 1.5% based on the value of load resis-tance (in the range 1–100 kO). The frequency of oscilla-tions at a wind speed of 10 mph and a load resistanceof 36.9 kO was measured to be 8.98 Hz. In comparison,the predicted frequency at 0 mph wind speed and at awind speed of 10 mph is 10.6 and 10.55 Hz, respec-tively. The decrease in frequency in the presence of inci-dent wind could indicate that apparent mass effects aresignificant and quasi-steady aerodynamics may not besufficient to represent the behavior at these operatingconditions. In a previous study of a galloping device

(Sirohi and Mahadik, 2011), the natural frequency ofthe device was lower and the predicted magnitude ofoutput power was closer to the measurements. Finally,the piezoelectric material constants that were used inthe analysis are valid only at small electric fields, assum-ing linear behavior. However, the piezoelectric materialshows significant nonlinearities at higher electric fieldssuch as those generated by the present device. For moreaccurate predictions, modified material constants willhave to be used (Sirohi and Chopra, 2000).

The correlation between measured and predictedpower over a range of incident wind velocities at a loadresistance of 36.9 kO is shown in Figure 17. The analy-sis captures the trend and magnitude except for thedrop in power at an incident wind velocity of 13.6 mph.

Conclusions and Future Work

A wind energy–harvesting device based on a gallopingbeam with piezoelectric sheets to convert mechanicalenergy into electrical energy has been fabricated, tested,and analyzed. The prototype device generated a maxi-mum output power of more than 50 mW at a windspeed of 11.6 mph. This power level is sufficient to sup-ply most of the commercially available wireless sensorscurrently in use and is significantly higher than thatachievable from vibrational energy harvesting. The mainadvantages of this device are its simplicity and robust-ness. In addition, the mechanism of energy harvestingfrom galloping is inherently self-limiting. Therefore, noadditional safety features need to be implemented tolimit the motion of the device under high winds.

A coupled electromechanical model of the system wasdeveloped using quasi-steady aerodynamics. The modelshowed good agreement with the experimental data overa range of operating conditions. There were some discre-pancies between the measurements and predictions, espe-cially in terms of the natural frequency and magnitudeof power output. It was concluded that at the naturalfrequency of the present system, approximately 10 Hz,apparent mass effects of the air could be significant. Inaddition, an abrupt decrease in output power at incidentwind velocities greater than 13.6 mph were observed.This was attributed to large-scale turbulence in the windtunnel and requires further investigation.

The galloping piezoelectric energy-harvesting devicehas been shown to be a viable alternative for poweringwireless sensor nodes. Future work will involve investi-gation of the fatigue properties of the device, responseto transient wind gusts, and refinements to the aerody-namic model.

Acknowledgments

The authors would like to thank the Cockrell School ofEngineering for providing funding for this research and Dr.Ashish Purekar for his constructive suggestions.

0

10

20

30

40

50

0 50 100 150 200 250 300

Load resistance (k")

Out

put p

ower

(mW

)

PredictedMeasured

Figure 16. Correlation between measured and predicted out-put power for an incident wind speed of 10 mph.

0

10

20

30

40

50

0 2 4 6 8 10 12

Wind velocity (mph)

Out

put p

ower

(mW

)

14

PredictedMeasured

Figure 17. Correlation between measured and predicted out-put power for a load resistance of 36.9 kO.


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