An experimentally validated bimorph cantilever model for ...

IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 18 (2009) 025009 (18pp) doi:10.1088/0964-1726/18/2/025009

An experimentally validated bimorphcantilever model for piezoelectric energyharvesting from base excitationsA Erturk1,3 and D J Inman2

1 Center for Intelligent Material Systems and Structures, Department of Engineering Scienceand Mechanics, Virginia Tech, Blacksburg, VA 24061, USA2 Center for Intelligent Material Systems and Structures, Department of MechanicalEngineering, Virginia Tech, Blacksburg, VA 24061, USA

E-mail: [email protected]

Received 28 May 2008, in final form 16 November 2008Published 13 January 2009Online at stacks.iop.org/SMS/18/025009

AbstractPiezoelectric transduction has received great attention for vibration-to-electric energyconversion over the last five years. A typical piezoelectric energy harvester is a unimorph or abimorph cantilever located on a vibrating host structure, to generate electrical energy from baseexcitations. Several authors have investigated modeling of cantilevered piezoelectric energyharvesters under base excitation. The existing mathematical modeling approaches range fromelementary single-degree-of-freedom models to approximate distributed parameter solutions inthe sense of Rayleigh–Ritz discretization as well as analytical solution attempts with certainsimplifications. Recently, the authors have presented the closed-form analytical solution for aunimorph cantilever under base excitation based on the Euler–Bernoulli beam assumptions. Inthis paper, the analytical solution is applied to bimorph cantilever configurations with series andparallel connections of piezoceramic layers. The base excitation is assumed to be translation inthe transverse direction with a superimposed small rotation. The closed-form steady stateresponse expressions are obtained for harmonic excitations at arbitrary frequencies, which arethen reduced to simple but accurate single-mode expressions for modal excitations. Theelectromechanical frequency response functions (FRFs) that relate the voltage output andvibration response to translational and rotational base accelerations are identified from themulti-mode and single-mode solutions. Experimental validation of the single-mode coupledvoltage output and vibration response expressions is presented for a bimorph cantilever with atip mass. It is observed that the closed-form single-mode FRFs obtained from the analyticalsolution can successfully predict the coupled system dynamics for a wide range of electricalload resistance. The performance of the bimorph device is analyzed extensively for the shortcircuit and open circuit resonance frequency excitations and the accuracy of the model is shownin all cases.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

The drastic reduction in power requirements of small electroniccomponents has motivated the research for powering suchcomponents by using the vibration energy available in their

3 Author to whom any correspondence should be addressed.

environment, especially in remote/wireless sensing applica-tions. As proposed by Williams and Yates [1], the threebasic vibration-to-electric energy conversion mechanisms areelectromagnetic [1–3], electrostatic [4] and piezoelectric [5–7]transductions. In the past decade, these transduction mech-anisms have been investigated by numerous researchers forvibration-based energy harvesting and extensive discussions

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Smart Mater. Struct. 18 (2009) 025009 A Erturk and D J Inman

can be found in the existing review articles (e.g., Beebyet al [8]). The literature of the last five years showsthat the piezoelectric transduction has received the greatestattention for vibration-to-electric energy conversion and threereview articles specifically dealing with piezoelectric energyharvesting have been published in the past two years [5–7].

Typically, a piezoelectric energy harvester is a cantileveredbeam with one or two piezoceramic layers (a unimorph ora bimorph). Basically, the harvester beam is located ona vibrating host structure and the dynamic strain inducedin the piezoceramic layer(s) generates an alternating voltageoutput across the electrodes covering the piezoceramiclayer(s). In addition to the experimental research onpossible applications of such harvesters, researchers haveproposed various mathematical models. Although theimplementation of piezoelectric energy harvesting for charginga real battery in an efficient way is more sophisticateddue to the AC-to-DC (alternating current-to-direct current)conversion process [9–13], researchers have considered aresistive electrical load in the circuit to come up with a simplemodel for predicting the electrical outputs for a given basemotion input. The coupled problem of predicting the voltageacross the resistive load connected to the electrodes of avibrating harvester under base excitation has been investigatedby many authors. The early modeling attempts of piezoelectricenergy harvesters employed single-degree-of-freedom (SDOF)solutions [14, 15]. SDOF modeling (i.e., lumped parametermodeling) is a convenient modeling approach since theelectrical domain already consists of lumped parameters: acapacitor (due to the internal capacitance of piezoceramic)and a resistor (due to an external load resistance). Hence,the only thing required is to obtain the lumped parametersrepresenting the mechanical domain so that the mechanicalequilibrium and electrical loop equations can be coupledthrough the piezoelectric constitutive relations [16]. Thiswas the main procedure followed by Roundy et al [14] andduToit et al [15] in their SDOF model derivations. AlthoughSDOF modeling gives initial insight into the problem byallowing simple expressions, it is an approximation limited toa single vibration mode and it lacks important aspects of thephysical system, such as the dynamic mode shape and accuratestrain distribution as well as their effects on the electricalresponse. Since cantilevered harvesters are excited due to themotion of their base, the well-known SDOF harmonic baseexcitation relation taken from the elementary vibration textshas been used in the energy harvesting literature both formodeling [15] and studying the optimization [17] of energyharvesters. It was recently shown [18] that the traditionalform of the SDOF harmonic base excitation relation mayyield highly inaccurate results both for the transverse andlongitudinal vibrations of cantilevered harvesters dependingon the tip (proof) mass to beam/bar mass ratio. Correctionfactors were derived [18] to improve the predictions of SDOFelectromechanical relations [15] of cantilevered harvestersunder base excitation.

As an improved modeling approach, the Rayleigh–Ritztype discrete formulation derived by Hagood et al [19] (basedon the generalized Hamilton’s principle for electromechanical

systems due to Crandall et al [20]) was employed by Sodanoet al [21] and duToit et al [15] for modeling of cantileveredpiezoelectric energy harvesters (based on the Euler–Bernoullibeam theory). The Rayleigh–Ritz solution gives a discretemodel of the distributed parameter system and it is a moreaccurate approximation compared to SDOF modeling. Inorder to represent the electrical outputs analytically, Lu et al[22] used the vibration mode shapes obtained from theEuler–Bernoulli beam theory and the piezoelectric constitutiverelation [16] that gives the electric displacement to relate theelectrical outputs to the mechanical mode shape. Similarmodels were given by Chen et al [23] and Lin et al [24]where the electrical response is expressed in terms of the beamvibration response. The issues in these analytical modelingattempts include not considering the resonance phenomenonand modal expansion as well as oversimplified modelingof piezoelectric coupling in the beam equation as viscousdamping [22–24]. As shown in this work, representing theeffect of piezoelectric coupling in the beam equation as viscousdamping fails in predicting the coupled system dynamics of apiezoelectric energy harvester, although this approach worksfor certain electromagnetic energy harvesters [1]. In termsof analytical modeling, more recently, Ajitsaria et al [25]presented a bimorph cantilever model, where they attempted tocombine the static sensing/actuation equations (with constantradius of curvature and a static tip force) with the dynamicEuler–Bernoulli beam equation (where the radius of curvaturevaries) under base excitation (where there is no tip force).Thus, highly different modeling approaches have appeared inthe literature during the past five years and some of themmight be misleading due to weak mathematical assumptionsinvolved [26].

Recently, Erturk and Inman [27] have presented theanalytical solution to the coupled problem of a unimorphpiezoelectric energy harvester configuration based on theEuler–Bernoulli assumptions. They obtained the coupledvoltage response across the resistive load and the coupledvibration response of the harvester explicitly for harmonicbase excitations in the form of translation with small rotation.The short circuit and open circuit trends and the effectof piezoelectric coupling were investigated extensively [27].Later, Elvin and Elvin [28] have observed the convergenceof the Rayleigh–Ritz type of solution formerly introduced byHagood et al [19] to the analytical solution given by Erturk andInman [27] when sufficient number of vibration modes is usedwith appropriate admissible functions.

This paper presents the application of the coupleddistributed parameter solution [27] to bimorph cantileverconfigurations with series and parallel connections ofpiezoceramic layers. The steady state voltage responseand vibration response expressions are derived for harmonicexcitation of the base at an arbitrary excitation frequency (inthe form of translation in the transverse direction with smallrotation). Then, by using the complete (multi-mode) solutions,the response expressions are reduced to simple but accuratesingle-mode relations. The single-mode relations can be usedinstead of the multi-mode relations for modal excitations (i.e.,for excitations around resonance) of cantilevered bimorphs

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Figure 1. Bimorph cantilever configurations with (a) series connection of piezoceramic layers, (b) parallel connection of piezoceramic layersand the (c) cross-sectional view of a bimorph cantilever.

since the resonance excitation is the main concern in vibration-based energy harvesting. The electromechanical FRFs thatgive the voltage output and vibration response-to-translationaland rotational base acceleration relations are extracted fromthe multi-mode and single-mode solutions. Experimentalvalidation of the analytical formulation is given for a bimorphcantilever with a tip mass. It is shown that the single-mode analytical relations proposed here are very accurate inpredicting the voltage output and vibration response FRFs. Thebimorph device is analyzed extensively for the short circuit andopen circuit resonance frequency excitations by using differentresistive loads and it is observed that the analytical model cansuccessfully predict the coupled system dynamics.

2. Fundamentals of the coupled distributedparameter model

This section reviews the assumptions in distributed parameterelectromechanical modeling and introduces the two possiblebimorph configurations based on the connection of thepiezoceramic layers. Derivation of the coupled beam equationin physical coordinates is given along with the relevantexpressions for the modal analysis. Derivation of theelectrical circuit equation for an instantaneous deflection of avibrating cantilever is explained based on the fundamentals ofpiezoelectricity and analytical structural dynamics.

2.1. Bimorph configurations and modeling assumptions

It is known from the literature of static sensing/actuationthat, depending on the voltage or current requirements, thepiezoceramic layers of a symmetric bimorph can be combinedin series or in parallel (see, for instance, Wang and Cross [29]).This common practice of static sensing/actuation problems isvalid for the dynamic piezoelectric energy harvesting problemas well. Each of the two bimorph configurations displayedin figures 1(a) and (b) undergoes bending vibrations due tothe motion of its base. The piezoceramic layers are assumedto be identical and conductive electrodes are assumed to befully covering the respective surfaces of these layers (top andbottom). The instantaneous bending strain in the top andbottom layers at an arbitrary position x over the beam lengthhave the opposite sign (i.e., one is in tension whereas the other

is in compression). As a consequence, since the piezoceramiclayers of the bimorph shown in figure 1(a) are poled oppositelyin the thickness direction (i.e., y-direction), this configurationrepresents the series connection of the piezoceramic layers.Likewise, figure 1(b) represents the parallel connection of thepiezoceramic layers because the layers are poled in the samedirection.

The bimorph cantilever configurations are modeled hereas uniform composite beams based on the Euler–Bernoullibeam assumptions. Therefore, plane sections are assumedto remain plane during the vibratory motion and the effectsof shear deformation and rotary inertia are neglected. Thisis a reasonable assumption since typical cantilevered energyharvesters are designed and manufactured as fairly thin beams.The mechanical losses are represented by internal and externaldamping mechanisms. The internal damping mechanism isassumed to be in the form of strain rate (or Kelvin–Voigt)damping and the effect of external (air) damping is consideredwith a separate damping coefficient. The piezoceramic andsubstructure layers are assumed to be perfectly bonded toeach other. The electrodes covering the opposite facesof piezoceramic layers are assumed to be very thin whencompared to the overall thicknesses of the harvester so thattheir contribution to the thickness dimension is negligible.

The continuous electrode pairs covering the top andthe bottom faces of the piezoceramic layers are assumedto be perfectly conductive so that a single electric potentialdifference can be defined across them. Therefore, theinstantaneous electric fields induced in the piezoceramic layersare assumed to be uniform throughout the length of the beam.A resistive electrical load (Rl) is considered in the circuitalong with the internal capacitances of the piezoceramic layers.Note that, considering a resistive load in the electrical domainis a common practice in modeling of vibration-based energyharvesters [14, 15, 21–28]. As a consequence, it is assumedthat the base motion input is persistent so that continuouselectrical outputs can be extracted from the electromechanicalsystem.

2.2. Coupled mechanical equation and modal analysis of abimorph cantilever

As far as the mechanical aspect of the problem is concerned,the bimorph configurations shown in figures 1(a) and (b) are

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identical. That is, they have the same geometric and materialproperties. However, the backward piezoelectric couplingeffect in the beam equation due to piezoelectric constitutiverelations is different for series and parallel connections of thepiezoceramic layers, and expectedly, this affects the vibrationresponse of the cantilever. In the following, the beam equationsare derived for these two configurations and the analyticalmodal analysis relations are presented.

The motion of the base for each of the cantilevers shownin figures 1(a) and (b) is represented by translation g(t) inthe transverse direction with superimposed small rotation h(t).Therefore, the effective base displacement wb(x, t) in thetransverse direction can be written as [27]

wb(x, t) = g(t)+ xh(t). (1)

The partial differential equation governing the forcedvibrations of a uniform cantilevered bimorph (with a tip mass)under base excitation is

∂2M(x, t)

∂x2+ cs I

∂5wrel(x, t)

∂x4 ∂ t+ ca

∂wrel(x, t)

∂ t

+ m∂2wrel(x, t)

∂ t2= −[m + Mtδ(x − L)]∂

2wb(x, t)

∂ t2(2)

wherewrel(x, t) is the transverse deflection of the beam relativeto its base at position x and time t , M(x, t) is the internalbending moment (excluding the strain rate damping effect), cs Iis the equivalent damping term of the composite cross-sectiondue to strain rate damping (cs is the equivalent coefficientof strain rate damping and I is the equivalent area momentof inertia of the composite cross-section), ca is the viscousair damping coefficient, m is the mass per unit length of thebeam, Mt it is tip mass and δ(x) is the Dirac delta function.Both of the damping mechanisms are assumed to satisfy theproportional damping criterion, hence, they are mathematicallyconvenient for the modal analysis solution procedure4. Notethat the effect of strain rate damping is an internal bendingmoment and it is directly written outside the term M(x, t) inequation (2).

Instead of defining the damping coefficients in thephysical equation of motion, one could consider thecorresponding undamped equation (by setting cs I = ca = 0in equation (2)) and introduce modal damping to the equationof motion in modal coordinates as is common practice. Itis worthwhile to mention that the foregoing consideration ofthe mechanical damping components results in an additionalexcitation term due to external damping as shown in Erturkand Inman [18]. Typically, for harvesters operating in air, theexternal damping excitation is negligible when compared to theinertial excitation term. It was shown in a dimensionless basisthat, in the absence of a tip mass, the amount of modal forcingdue to external damping term is less than 5% of the total modalbase excitation force if the component of the modal dampingratio due to external damping is less than 2.5% (see figure 3

4 Strain rate damping is assumed to be stiffness proportional whereas airdamping is assumed to be mass proportional and this type of damping is alsoknown as the Rayleigh damping [30]. Modeling and identification of moresophisticated damping mechanisms in beams were investigated by Banks andInman [31].

in [18]). Therefore the damping excitation term is directlyomitted in equation (2) for simplicity. However, excitation dueto external damping can be important for harvesters operatingin fluids with larger damping effect and the general form of theforcing function must be used in that case [18, 27].

The internal bending moment term in equation (2) is thefirst moment of axial strain over the cross-section:

M(x, t) = −b

(∫ −h s/2

−hp−h s/2T p

1 y dy +∫ h s/2

−h s/2T s

1 y dy

+∫ hp+h s/2

h s/2T p

1 y dy

)(3)

where b is the width, h p is the thickness of each piezoceramiclayer and h s is the thickness of the substructure layer(figure 1(c)). Furthermore, T p

1 and T s1 are the axial stress

components in the piezoceramic and substructure layers,respectively (1-direction is the longitudinal direction, i.e., x-direction), and they are given by the following constitutiverelations:

T s1 = YsS s

1, T p1 = cE

11Sp1 − e31 E3 (4)

where Ys is Young’s modulus of the substructure layer, cE11 is

the elastic stiffness (i.e., Young’s modulus) of the piezoceramiclayer at constant electric field, e31 is the piezoelectric constantand E3 is the electric field component in 3-direction (i.e., y-direction). Here and hereafter, the subscripts and superscriptsp and s stand for the piezoceramic and the substructure layers,respectively. Based on the plane-stress assumption for a beam,the elastic stiffness component can be expressed as cE

11 =1/sE

11, where sE11 is the elastic compliance at constant electric

field. Furthermore, based on the same assumption, e31 canbe given in terms of the more commonly used piezoelectricconstant d31 as e31 = d31/sE

11. The axial strain components in

the piezoelectric and substructure layers are given by Sp1 and

S s1, respectively, and they are due to bending only. Hence the

axial strain at a certain level (y) from the neutral axis of thecomposite beam is simply proportional to the curvature of thebeam at that position (x):

S1(x, y, t) = −y∂2wrel(x, t)

∂x2. (5)

The electric field component E3 should be expressedin terms of the respective voltage term in each bimorphconfiguration (figures 1(a) and (b)). This is the point wherethe resulting mechanical equations for series and parallelconnections of the piezoceramic layers differ from each other.Since the piezoceramic layers are assumed to be identical,voltage across the electrodes of each piezoceramic layer isvs(t)/2 in the series connection case (figure 1(a)). Expectedly,for the parallel connection case (figure 1(b)), voltage across theelectrodes of each piezoceramic layer is vp(t). It is worthwhileto add that e31 has the opposite sign for the top and thebottom piezoceramic layers for the series connection case (dueto opposite poling) so that the instantaneous electric fieldsare in the same direction (i.e., E3(t) = −vs(t)/2h p in bothlayers). For the configuration with parallel connection, sincee31 has the same sign in top and bottom piezoceramic layers,

4


the instantaneous electric fields are in the opposite directions(i.e., E3(t) = −vp(t)/h p in the top layer and E3(t) =vp(t)/h p in the bottom layer). Another important point isthat, for both configurations, the piezoelectric coupling termcoming from equation (3) is a function of time only. Hence,before substituting equation (3) into (2), the electrical termmust be multiplied by [H (x)− H (x − L)], where H (x) isthe Heaviside function. Since the voltage outputs of the seriesand parallel connection cases are different, the piezoelectriccoupling effect in the mechanical equation (equation (2))is expected to be different. Thus, in the rest of thepaper, the mechanical response expressions of the series andparallel connection configurations are denoted by ws

rel(x, t)and wp

rel(x, t), respectively. Note that, here and hereafter, thesubscripts and superscripts s and p stand for series and parallelconnections of the piezoceramic layers.

Based on the foregoing discussion, the coupled beamequation can be obtained for the series connection case(figure 1(a)) as follows:

Y I∂4ws

rel(x, t)

∂x4+ cs I

∂5wsrel(x, t)

∂x4∂ t+ ca

∂wsrel(x, t)

∂ t

+ m∂2ws

rel(x, t)

∂ t2+ ϑsvs(t)

[dδ(x)

dx− dδ(x − L)

dx

]

= −[m + Mtδ(x − L)]∂2wb(x, t)

∂ t2(6)

where the piezoelectric coupling term ϑs for the seriesconnection case is

ϑs = e31b

2h p

[h2

s

4−

(h p + h s

2

)2]. (7)

Similarly, one can obtain the equation of motion for thecase with the parallel connection of the piezoceramic layers as(figure 1(b))

Y I∂4w

prel(x, t)

∂x4+ cs I

∂5wprel(x, t)

∂x4∂ t+ ca

∂wprel(x, t)

∂ t

+ m∂2w

prel(x, t)

∂ t2+ ϑpvp(t)

[dδ(x)

dx− dδ(x − L)

dx

]

= −[m + Mtδ(x − L)]∂2wb(x, t)

∂ t2(8)

where the backward coupling term ϑp for the parallelconnection case can be expressed as

ϑp = 2ϑs = e31b

h p

[h2

s

4−

(h p + h s

2

)2]. (9)

In equations (6) and (8), the bending stiffness term Y I andthe mass per unit length term m are simply

Y I = 2b

3

[Ys

h3s

8+ cE

11

((h p + h s

2

)3

− h3s

8

)],

m = b(ρsh s + 2ρph p)

(10)

where ρs and ρp are the mass densities of the substructure andthe piezoceramic materials, respectively.

Based on the proportional damping assumption, thevibration response relative to the base of the bimorph(figures 1(a) and (b)) can be represented as an absolutely anduniformly convergent series of the eigenfunctions as

wsrel(x, t) =

∞∑r=1

φr (x)ηsr (t), (11a)

wprel(x, t) =

∞∑r=1

φr (x)ηpr (t) (11b)

where φr (x) is the mass normalized eigenfunction of the r thvibration mode, ηs

r (t) and ηpr (t) are the modal mechanical

response expressions of the series and parallel connectioncases, respectively. The eigenfunctions denoted by φr (x)are the mass normalized eigenfunctions of the correspondingundamped free vibration problem:

φr (x) = Cr

[cos

λr

Lx − cosh

λr

Lx +ςr

(sin

λr

Lx − sinh

λr

Lx

)]

(12)where ςr is obtained from

ςr = sin λr − sinhλr + λrMtmL (cos λr − cosh λr )

cos λr + cosh λr − λrMtmL (sinλr − sinhλr )

(13)

and Cr is a modal amplitude constant which should beevaluated by normalizing the eigenfunctions according to thefollowing orthogonality conditions:∫ L

0φs(x)mφr(x) dx + φs(L)Mtφr (L)

+[

dφs(x)

dxIt

dφr (x)

dx

]x=L

= δrs

∫ L

0φs(x)Y I

d4φr (x)

dx4dx −

[φs(x)Y I

d3φr (x)

dx3

]x=L

+[

dφs(x)

dxY I

d2φr (x)

dx2

]x=L

= ω2r δrs .

(14)

Here, It is the rotary inertia of the tip mass Mt and δrs isKronecker delta, defined as being equal to unity for s = r andequal to zero for s �= r . Furthermore, ωr is the undampednatural frequency of the r th vibration mode in short circuitconditions (i.e., as Rl → 0) given by

ωr = λ2r

√Y I

mL4(15)

where the eigenvalues of the system (λr for mode r ) areobtained from

1 + cos λ cosh λ+ λMt

mL(cosλ sinh λ− sinλ cosh λ)

− λ3 It

mL3(cosh λ sinλ+ sinh λ cos λ)

+ λ4 Mt It

m2 L4(1 − cos λ cosh λ) = 0. (16)

It should be mentioned that the foregoing modal analysisis given for the short circuit conditions (i.e., for Rl → 0) so

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(a) (b)

Figure 2. (a) Cantilever beam with a single piezoceramic layer under transverse vibrations (exaggerated view) and the (b) correspondingelectrical circuit for a resistive electrical load connected to the electrodes.

that the conventional form of the eigenfunctions is obtainedin equation (12) (since, for short circuit conditions, vs(t) →0 and vp(t) → 0 in equations (6) and (8), respectively).Thus, for a given bimorph, the form of the eigenfunctionsgiven by φr (x) and their mass normalization conditions arethe same regardless of the series or parallel connections of thepiezoceramic layers. For non-zero values of load resistance,the voltage terms in the mechanical equations take finite values,generating point moment excitations at the boundaries of thepiezoceramic layer according to equations (6) and (8), andyielding two different modal mechanical response functions forthese equations as ηs

r (t) and ηpr (t), respectively (as obtained in

sections 3.3 and 4.3). Therefore, the feedback from the voltageresponse for a given load resistance alters the mechanicalresponse as well as the resonance frequency of the harvester,which are observed experimentally and predicted theoreticallyin section 7. At this stage, it should be underlined thatthe harvester beam has the resonance characteristics of thecorresponding uncoupled (or passive) beam for Rl → 0 only.

2.3. Coupled electrical circuit equation of a piezoceramiclayer under dynamic bending

In order to derive the governing circuit equations of thebimorph configurations for series and parallel connectionsof the piezoceramic layers, one should first examine theelectrical dynamics of a single layer under bending vibrations.Figure 2(a) displays a cantilevered beam with a singlepiezoceramic layer, i.e., a unimorph cantilever. Note that thedeflections are exaggerated to highlight the space- and time-dependent radius of curvature of the neutral axis at an arbitrarypoint. The electrodes bracketing the piezoceramic layers fullycover the top and the bottom surfaces and they are connectedto a resistive electrical load.

Since the only source of mechanical strain is assumed tobe the axial strain due to bending, the tensorial representationof the relevant piezoelectric constitutive relation [16] that givesthe vector of electric displacements can be reduced to thefollowing scalar equation:

D3 = e31Sp1 + εS

33 E3 (17)

where D3 is the electric displacement component and εS33 is the

permittivity component at constant strain with the plane-stressassumption (εS

33 = εT33 − d2

31/sE11 where εT

33 is the permittivitycomponent at constant stress). Since the circuit admittance

across the electrodes is 1/Rl, the electric current output canbe obtained from the Gauss law as [16]

d

dt

(∫A

D · n dA

)= v(t)

Rl(18)

where D is the vector of electric displacement componentsin the piezoceramic layer, n is the unit outward normaland the integration is performed over the electrode areaA [16, 27]. As can be anticipated, the only contribution tothe inner product of the integrand in equation (18) is from D3,since the electrodes are perpendicular to 3-direction (i.e., y-direction). After expressing the average bending strain in thepiezoceramic layer in terms of the curvature (see equation (5))and the uniform electric field in terms of the electric potentialdifference (E3(t) = −v(t)/h p), equation (17) can be used inequation (18) to obtain

εS33bL

h p

dv(t)

dt+ v(t)

Rl= −e31h pcb

∫ L

0

∂3wrel(x, t)

∂x2∂ tdx (19)

where b, h p and L are the width, thickness and the length ofthe piezoceramic layer, respectively, and h pc is the distancebetween the neutral axis and the center of the piezoceramiclayer [27]. One can then substitute the modal expansion formgiven by

wrel(x, t) =∞∑

r=1

φr (x)ηr (t) (20)

in equation (19) to obtain

εS33bL

h p

dv (t)

dt+ v(t)

Rl=

∞∑r=1

κrdηr (t)

dt(21)

where κr is the modal coupling term in the electrical circuitequation:

κr = −e31h pcb∫ L

0

d2φr (x)

dx2dx = −e31h pcb

dφr (x)

dx

∣∣∣∣x=L

.

(22)The forward coupling term κr has important consequences

as discussed by Erturk et al [27, 32] extensively. Accordingto equation (19), which originates from the Gauss lawgiven by equation (18), the excitation of the simple RCcircuit considered here as well as that of more sophisticatedharvesting circuit topologies [9–13] is proportional to theintegral of the dynamic strain distribution over the electrodearea. For vibration modes of a cantilevered beam other than

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the fundamental (first) mode, the dynamic strain distributionover the beam length changes sign at the strain nodes. It isknown from equation (5) that the curvature at a point is a directmeasure of the bending strain. Hence, for modal excitations,strain nodes are the inflection points of the eigenfunctions andthe integrand in equation (22) is the curvature eigenfunction.If the electric charge developed at the opposite sides ofa strain node is collected by continuous electrodes forvibrations with a certain mode shape, cancelation occurs dueto the phase difference in the mechanical strain distribution.Mathematically, the partial areas under the integrand functionof the integral in equation (22) cancel each other overthe domain of integration. As an undesired consequence,the excitation of the electrical circuit, and therefore theelectrical outputs may diminish drastically. In order to avoidcancellations, segmented electrodes can be used in harvestingenergy from the modes higher than the fundamental mode.The leads of the segmented electrodes can be combined inthe circuit in an appropriate manner [32]. Note that the r thvibration mode of a clamped-free beam has r − 1 strain nodes,and consequently, the first mode of a cantilevered beam hasno cancelation problem. Some boundary conditions are moreprone to strong cancellations. For instance, a beam withclamped–clamped boundary conditions has r + 1 strain nodesfor the r th vibration mode.

Based on equation (21), it is very useful to representthe electrical domain of the coupled system by the simplecircuit shown in figure 2(b). It is known in the circuitry-basedenergy harvesting literature that a piezoelectric element canbe represented as a current source in parallel with its internalcapacitance [9, 10]. Therefore, the simple circuit shown infigure 2(b) is the complete circuit of the electrical domainfor a single resistive load case. Note that, this representationconsiders the electrical domain only and the electromechanicalrepresentation of the coupled system is actually a transformerbecause of the voltage feedback sent to the mechanical domaindue to piezoelectric coupling (which will be incorporated inthe formulation here). The components of the circuit are theinternal capacitance Cp of the piezoceramic layer, the resistiveload Rl and the current source ip(t). In agreement withfigure 2(a), the voltage across the resistive load is denoted byv(t). Then, the Kirchhoff laws can be applied to the electricalcircuit shown in figure 2(b) to obtain

Cpdv(t)

dt+ v(t)

Rl= ip(t) (23)

where the internal capacitance and the current source terms canbe extracted by matching equations (21) and (23) as

Cp = εS33bL

h p, ip(t) =

∞∑r=1

κrdηr (t)

dt. (24)

Identification of the above terms (especially the currentsource term) has a very practical use for modeling ofmulti-morph harvesters. This way, for a given number ofpiezoceramic layers, there is no need to derive the electricalcircuit equation by using the constitutive relation and the Gausslaw given by equations (17) and (18), respectively. Each

piezoceramic layer will have a similar capacitance and currentsource term and the layers can be combined to the resistiveelectrical load(s) in a desired way. Here, however, we limitour discussion to bimorphs (two piezoceramic layers only) aspresented in the following.

3. Bimorph cantilever model for series connection ofthe piezoceramic layers

Based on the fundamentals given in section 2, this sectionpresents the derivation of the closed-form expressions forthe coupled voltage response vs(t) and vibration responsews

rel(x, t) of the bimorph configuration shown in figure 1(a).First the coupled mechanical equation is given in modalcoordinates and then the coupled circuit equation is derived.The resulting coupled equations are then solved for the steadystate voltage response and vibration response for harmonicbase motion inputs.

3.1. Coupled beam equation in modal coordinates

After substituting equation (11a) into (6) and applyingthe orthogonality conditions given by equation (14), themechanical equation of motion in modal coordinates can beobtained as

d2ηsr (t)

dt2+ 2ζrωr

dηsr (t)

dt+ ω2

r ηsr (t)+ χ s

r vs(t) = fr (t) (25)

where the modal electromechanical coupling term is

χ sr = ϑs

dφr (x)

dx

∣∣∣∣x=L

(26)

and the modal mechanical forcing function can be expressed as

fr (t) = −m

(d2g(t)

dt2

∫ L

0φr (x) dx + d2h (t)

dt2

∫ L

0xφr(x) dx

)

− Mtφr (L)

(d2g(t)

dt2+ L

d2h(t)

dt2

). (27)

In equation (25), ζr is the modal mechanical dampingratio that includes the combined effects of strain rate and airdamping. In the absence of a tip mass, how to relate the modaldamping ratio to the strain rate and air damping terms cs Iand ca mathematically based on the assumption of proportionaldamping can be found in the literature [18]. However, asa common experimental modal analysis practice, one canidentify the modal damping ratio ζr of a desired mode directlyfrom the frequency response or time domain measurements. Inthis way, the requirement of defining and obtaining the physicaldamping terms cs I and ca is avoided [27].

3.2. Coupled electrical circuit equation

As described in section 2.1, the piezoceramic layers of thebimorph configuration shown in figure 1(a) are connected inseries. We know from the practice given in section 2.3 thateach piezoceramic layer can be represented as a current sourcein parallel with its internal capacitance. Therefore, figure 3

7


Figure 3. Electrical circuit representing the series connection of thepiezoceramic layers.

displays the series connection of the identical piezoceramiclayers of the bimorph configuration shown in figure 1(a).

Kirchhoff laws can be applied to the circuit depicted infigure 3 to obtain

Cp

2

dvs(t)

dt+ vs(t)

Rl= i s

p(t) (28)

where the internal capacitance and the current source terms ofthe bimorph (for each layer) are

Cp = εS33bL

h p, i s

p(t) =∞∑

r=1

κrdηs

r (t)

dt. (29)

The modal coupling term is then

κr = −e31h pcb∫ L

0

d2φr (x)

dx2dx

= − e31(h p + h s)b

2

dφr (x)

dx

∣∣∣∣x=L

(30)

where h pc (the distance between the neutral axis and thecenter of the piezoceramic layer) is expressed in terms of thepiezoceramic and the substructure layer thicknesses h p andh s (figure 1(c)). Hence, equation (28) is the electrical circuitequation of the bimorph cantilever for series connection of thepiezoceramic layers.

3.3. Closed-form voltage response and vibration responseexpressions

Equations (25) and (28) constitute the coupled equations forthe modal mechanical response ηs

r (t) of the bimorph and thevoltage response vs(t) across the resistive load. In this section,we derive the steady state solution of these terms for harmonicmotion inputs. If the translational and rotational componentsof the base displacement given by equation (1) are harmonic ofthe forms g(t) = Y0ejωt and h(t) = θ0ejωt , where Y0 and θ0 arethe translational and small rotational displacement amplitudesof the base, ω is the frequency and j is the unit imaginarynumber, then the modal forcing function given by equation (27)can be expressed as fr (t) = Fr ejωt where the amplitude Fr is

Fr = ω2

[m

(Y0

∫ L

0φr (x)dx + θ0

∫ L

0xφr (x)dx

)

+ Mtφr (L)(Y0 + Lθ0)

]. (31)

For the harmonic base motions at frequency ω, the steadystate modal mechanical response of the beam and the steady

state voltage response across the resistive load are assumedto be harmonic at the same frequency as ηs

r (t) = H sr ejωt and

vs(t) = Vsejωt (linear system assumption), respectively, wherethe amplitudes H s

r and Vs are complex valued. Therefore,equations (25) and (28) yield the following two equations forH s

r and Vs:

(ω2r − ω2 + j2ζrωrω)H

sr + χ s

r Vs = Fr (32)(

1

Rl+ jω

Cp

2

)Vs − jω

∞∑r=1

κr H sr = 0. (33)

The complex modal mechanical response amplitude H sr

can be extracted from equation (32) and it can be substitutedin equation (33) to obtain the complex voltage amplitude Vs

explicitly. The resulting complex voltage amplitude can thenbe used in vs(t) = Vsejωt to express the steady state voltageresponse as

vs(t) =∑∞

r=1jωκr Fr

ω2r −ω2+j2ζrωrω

1Rl

+ jωCp

2 + ∑∞r=1

jωκrχsr


ejωt . (34)

The complex voltage amplitude Vs can be substituted intoequation (32) to obtain the steady state modal mechanicalresponse of the bimorph as

ηsr (t) =

(Fr − χ s

r

∑∞r=1

jωκr Fr


1Rl

+ jωCp

2 + ∑∞r=1

jωκrχsr


)

× ejωt

ω2r − ω2 + j2ζrωrω

. (35)

The transverse displacement response (relative to thebase) at point x on the bimorph can be obtained in physicalcoordinates by substituting equation (35) in equation (11a):

wsrel(x, t)

=∞∑

r=1

[(Fr − χ s

r

∑∞r=1

jωκr Fr


1Rl

+ jωCp

2 + ∑∞r=1

jωκrχsr


)

× φr (x)ejωt


]. (36)

Note that the vibration response given by equation (36)is the response of the beam relative to its moving base. Ifone is interested in the coupled beam displacement in theabsolute physical coordinates (relative to the fixed frame), itis the superposition of the base displacement and the vibratorydisplacement relative to base:

ws(x, t) = wb(x, t)+wsrel(x, t) (37)

where wb(x, t) is the base displacement given by equation (1).

4. Bimorph cantilever model for parallel connectionof the piezoceramic layers

This section aims to derive the steady state expressions forvoltage response vp(t) and the vibration response wp

rel(x, t) ofthe bimorph configuration shown in figure 1(b) to harmonicbase motions. The coupled beam equation in modalcoordinates and the electrical circuit equations are derived andthe closed-form solutions are obtained in the following.

8


Figure 4. Electrical circuit representing the parallel connection ofthe piezoceramic layers.

4.1. Coupled beam equation in modal coordinates

After substituting equation (11b) in equation (8), the partialdifferential equation given by equation (8) can be reducedto an infinite set of ordinary differential equations in modalcoordinates as follows:

d2ηpr (t)

dt2+ 2ζrωr

dηpr (t)

dt+ ω2

r ηpr (t)+ χ p

r vp(t) = fr (t) (38)

where the modal electromechanical coupling term is

χ pr = ϑp

dφr (x)

dx

∣∣∣∣x=L

(39)

and the modal mechanical forcing function is given byequation (27). The discussion regarding the mechanicaldamping ratio ζr is the same as given in section 3.1.Thus, equation (38) is the coupled beam equation in modalcoordinates for the bimorph configuration with parallelconnection of the piezoceramic layers.

4.2. Coupled electrical circuit equation

It was mentioned in section 2.1 that the piezoceramic layers ofthe bimorph configuration shown in figure 1(b) are connectedin parallel. Since each of the piezoceramic layers can berepresented as a current source in parallel with its internalcapacitance (section 2.3), figure 4 represents the parallelconnection of the identical top and bottom piezoceramic layersof the bimorph configuration shown in figure 1(b).

One can then derive the governing circuit equation basedon the Kirchhoff laws as follows:

Cpdvp(t)

dt+ vp(t)

2Rl= i p

p (t) (40)

where the internal capacitance and the current source terms foreach layer are

Cp = εS33bL

h p, i p

p(t) =∞∑

r=1

κrdηp

r (t)

dt(41)

and the modal coupling term κr is given by equation (30).Equation (40) is the electrical circuit equation of the bimorphcantilever for parallel connection of the piezoceramic layers.

4.3. Closed-form voltage response and vibration responseexpressions

In order to solve for ηpr (t) and vp(t) in equations (38) and (40),

we follow the same procedure given in section 3.3 by assumingthe base excitation components in figure 1(b) to be harmonicas g(t) = Y0ejωt and h(t) = θ0ejωt . For these harmonic

base motion inputs of the same frequency, the modal forcingis harmonic as fr (t) = Fr ejωt where the amplitude Fr is givenby equation (31).

Based on the linear system assumption, the modalmechanical response ηp

r (t) and the voltage response vp(t) areassumed to be harmonic at the frequency of excitation such thatη

pr (t) = H p

r ejωt and vp(t) = Vpejωt , where the amplitudes H pr

and Vp are complex valued. Hence, equations (38) and (40)yield the following equations for H p

r and Vp:

(ω2r − ω2 + j2ζrωrω)H

pr + χ p

r Vp = Fr (42)

(1

2Rl+ jωCp

)Vp − jω

∞∑r=1

κr H pr = 0 (43)

where H pr and Vp can be obtained explicitly. Using the

resulting complex voltage amplitude in vp(t) = Vpejωt givesthe steady state voltage response as

vp(t) =∑∞

r=1jωκr Fr


12Rl

+ jωCp + ∑∞r=1

jωκrχp

r


ejωt . (44)

Then the steady state modal mechanical response of thebimorph can be obtained by using Vp in equation (42) as

ηpr (t) =

(Fr − χ p

r

∑∞r=1

jωκr Fr


12Rl

+ jωCp + ∑∞r=1

jωκrχp

r


)

× ejωt


. (45)

The modal mechanical response expression can then beused in equation (11b) to obtain the transverse displacementresponse (relative to the base) at point x on the bimorph:

wprel(x, t)

=∞∑

r=1

[(Fr − χ p

r

∑∞r=1

jωκr Fr


12Rl

+ jωCp + ∑∞r=1

jωκrχp

r


)

× φr (x)ejωt


]. (46)

Having obtained the vibration response relative to themoving base, one can easily use superpose the base motionto the relative response to obtain the transverse displacementresponse at point x relative to the fixed frame as follows:

wp(x, t) = wb(x, t)+ wprel(x, t) (47)

where the base displacement wb(x, t) is given by equation (1).

5. Single-mode electromechanical expressions formodal excitations

The steady state voltage response and vibration responseexpressions obtained in sections 2 and 3 are valid for harmonicexcitations at any arbitrary frequency ω. That is, equations (34)and (36) for series connection of the piezoceramic layers(figure 1(a)) and equations (44) and (46) for parallel connectionof the piezoceramic layers (figure 1(b)) are the multi-mode

9


solutions as they include all vibration modes of the bimorphharvester. Hence, these equations can predict the coupledsystem dynamics not only for resonance excitations but also forexcitations at the off-resonance frequencies of the harvester.

In order to obtain the maximum electrical response, itis preferable to excite a given harvester at its fundamentalresonance frequency (or at one of the higher resonancefrequencies). Most of the studies in the literature have focusedon the resonance excitation at the fundamental resonancefrequency in order to investigate the maximum performanceof the harvester for electrical power generation. Consequently,excitation of a bimorph at or very close to one of its naturalfrequencies is a very useful problem to investigate through theresulting equations derived in sections 3 and 4. This is themodal excitation condition and mathematically it correspondsto ω ∼= ωr . With this assumption on the excitationfrequency, the major contribution in the summation terms ofequations (34), (36), (44) and (46) are from the r th vibrationmode, which allows drastic simplifications in the coupledvoltage and vibration response expressions. In the following,the reduced single-mode expressions are given for excitationsat or very close to the r th natural frequency, however, it shouldbe noted that the fundamental mode is the main concern in theenergy harvesting problem (which corresponds to r = 1).

5.1. Series connection of the piezoceramic layers

If the bimorph configuration shown in figure 1(a) is excited atω ∼= ωr , the contribution of all the vibration modes other thanthe r th mode can be ignored in the summation terms. Then,the steady state voltage response given by equation (34) can bereduced to

vs(t) = j2ωRlκr Fr ejωt

(2 + jωRlCp)(ω2r − ω2 + j2ζrωrω)+ j2ωRlκrχ s

r(48)

and the transverse displacement relative to the moving base issimply obtained from equation (36) as

wsrel(x, t)

= (2 + jωRlCp)Frφr (x)ejωt


r

(49)

where the relevant terms can be found in section 3. Here andbelow, a hat (ˆ) denotes that the respective term is reducedfrom the full solution for excitations very close to a naturalfrequency.

5.2. Parallel connection of the piezoceramic layers

Similarly, if the bimorph configuration displayed in figure 1(b)is excited at ω ∼= ωr , the steady state voltage response givenby equation (44) can be reduced to

vp(t)

= j2ωRlκr Fr ejωt

(1 + j2ωRlCp)(ω2r − ω2 + j2ζrωrω)+ j2ωRlκrχ

pr

(50)

and the transverse displacement relative to the base is obtainedfrom equation (46) as

wprel(x, t)

= (1 + j2ωRlCp)Frφr (x)ejωt


pr

(51)

where the relevant terms can be found in section 4.

6. Multi-mode and single-mode electromechanicalFRFs

In the electromechanical model proposed, the two excitationinputs to the system are the translation of the base inthe transverse direction and its small rotation (figures 1(a)and (b)). For these two inputs, the resulting electricaloutputs are the voltage response and the vibration response.Therefore, for harmonic base excitations, one can definefour electromechanical FRFs between these two outputs andtwo inputs: voltage output-to-translational base acceleration,voltage output-to-rotational base acceleration, vibrationresponse-to-translational base acceleration and vibrationresponse-to-rotational base acceleration. This section extractsthese FRFs from the multi-mode (for arbitrary frequencyexcitations) and single-mode (for modal excitations) solutionsderived in the previous sections.

6.1. Multi-mode electromechanical FRFs

Since the translation and small rotation of the base are givenby g(t) = Y0ejωt and h(t) = θ0ejωt , the modal forcingfunction is in the form of fr (t) = Fr ejωt where Fr is givenby equation (31). Before identifying the aforementionedFRFs, one should first rearrange the complex modal forcingamplitude given by equation (31) as follows:

Fr = −σrω2Y0 − τrω

2θ0 (52)

where

σr = −m∫ L

0φr (x) dx − Mtφr (L) (53)

τr = −m∫ L

0xφr (x) dx − Mt Lφr (L). (54)

6.1.1. Series connection of the piezoceramic layers. Thesteady state voltage response given by equation (34) canbe written in terms of the translational and rotational baseaccelerations as

vs(t) = αs(ω)(−ω2Y0ejωt )+ μs(ω)(−ω2θ0ejωt ) (55)

where the FRF that relates the voltage output to translationalbase acceleration is

αs(ω) =∑∞

r=1jωκrσr


1Rl

+ jωCp

2 + ∑∞r=1

jωκrχ sr


(56)

10


and the voltage output per rotational base acceleration inputcan be given by

μs(ω) =∑∞

r=1jωκr τr


1Rl

+ jωCp

2 + ∑∞r=1

jωκrχsr


. (57)

Similarly, the steady state vibration response relative to thebase of the bimorph given by equation (36) can be expressedas

wsrel(x, t) = βs(ω, x)(−ω2Y0ejωt )+ ψs(ω, x)(−ω2θ0ejωt )

(58)where the transverse displacement response-to-translationalbase acceleration FRF is

βs(ω, x)

=∞∑

r=1

[(σr − χ s

r

∑∞r=1

jωκrσr


1Rl

+ jωCp

2 + ∑∞r=1

jωκrχsr


)

× φr (x)


](59)

and the transverse displacement response and rotational baseacceleration are related by

ψs(ω, x)

=∞∑

r=1

[(τr − χ s

r

∑∞r=1

jωκr τr


1Rl

+ jωCp

2 + ∑∞r=1

jωκrχsr


)

× φr (x)


]. (60)

6.1.2. Parallel connection of the piezoceramic layers. It ispossible to derive similar FRFs for the parallel connection ofthe piezoceramic layers. The steady state voltage responsegiven by equation (44) can be rearranged to give

vp(t) = αp(ω)(−ω2Y0ejωt

) + μp(ω)(−ω2θ0ejωt ) (61)

where the voltage output-to-translational base accelerationFRF is

αp(ω) =∑∞

r=1jωκrσr


12Rl

+ jωCp + ∑∞r=1

jωκrχp

r


(62)

and the voltage output-to-rotational base acceleration FRF canbe given by

μp(ω) =∑∞

r=1jωκr τr


12Rl

+ jωCp + ∑∞r=1

jωκrχp

r


. (63)

From equation (46), the steady state vibration responserelative to the base of the bimorph can be expressed as

wprel(x, t) = βp(ω, x)(−ω2Y0ejωt )+ ψp(ω, x)(−ω2θ0ejωt )

(64)

where the transverse displacement response-to-translationalbase acceleration FRF is

βp(ω, x)

=∞∑

r=1

[(σr − χ p

r

∑∞r=1

jωκrσr


12Rl

+ jωCp + ∑∞r=1

jωκrχp

r


)

× φr (x)


](65)

and the transverse displacement response-to-rotational baseacceleration FRF is

ψp(ω, x)

=∞∑

r=1

[(τr − χ p

r

∑∞r=1

jωκr τr


12Rl

+ jωCp + ∑∞r=1

jωκrχp

r


)

× φr (x)


]. (66)

6.2. Single-mode electromechanical FRFs

In order to extract the respective FRFs of the single-modeexpressions, one should use equations (48)–(51) along withequation (52). In the following, equation (52) is substituted ineach of equations (48)–(51) and the relevant FRFs are extractedas done for the multi-mode solution case. Note that, the single-mode electromechanical FRFs given here are strictly valid formodal excitations (ω ∼= ωr ) only.

6.2.1. Series connection of the piezoceramic layers.Equation (48) can be rearranged to give the single-mode steadystate voltage response as

vs(t) = αs(ω)(−ω2Y0ejωt )+ μs(ω)(−ω2θ0ejωt ) (67)

where the single-mode FRF that relates the voltage output totranslational base acceleration is

αs(ω) = j2ωRlκrσr


r(68)

and the single-mode voltage output-to-rotational base acceler-ation FRF is

μs(ω) = j2ωRlκrτr


r

.

(69)The single-mode steady state vibration response relative

to the base of the bimorph given by equation (49) can berearranged to give

wsrel(x, t) = βs(ω, x)(−ω2Y0ejωt )+ ψs(ω, x)(−ω2θ0ejωt )

(70)where the single-mode transverse displacement response-to-translational base acceleration FRF is

βs(ω, x)

= (2 + jωRlCp)σrφr (x)


r

(71)

11


and the single-mode transverse displacement response-to-rotational base acceleration FRF can be given by

ψs(ω, x)

= (2 + jωRlCp)τrφr (x)


r

. (72)

6.2.2. Parallel connection of the piezoceramic layers.The single-mode steady state voltage response given byequation (50) can be expressed in terms of the translational androtational base accelerations as

vp(t) = αp(ω)(−ω2Y0ejωt )+ μp(ω)(−ω2θ0ejωt ) (73)

where the single-mode FRF that relates the voltage output totranslational base acceleration is

αp(ω)

= j2ωRlκrσr


pr

(74)

and the single-mode FRF that relates the voltage output torotational base acceleration is

μp(ω)

= j2ωRlκrτr


pr.

(75)

Similarly, the single-mode steady state vibration responserelative to the base of the bimorph given by equation (51) canbe rewritten as

wprel(x, t) = βp(ω, x)(−ω2Y0ejωt )+ ψp(ω, x)(−ω2θ0ejωt )

(76)where the single-mode transverse displacement response-to-translational base acceleration FRF can be given by

βp(ω, x)

= (1 + j2ωRlCp)σrφr (x)


pr

(77)

and the single-mode transverse displacement response-to-rotational base acceleration FRF is

ψp(ω, x)

= (1 + j2ωRlCp)τrφr (x)


pr.

(78)

7. Experimental validation

This section provides experimental validation of the single-mode analytical relationships. The experimentally measuredvoltage response-to-base acceleration FRFs and the vibrationresponse-to-base acceleration FRFs are compared with theclosed-form FRFs derived in this paper. Variations ofthe voltage output and the tip velocity response of thebimorph with changing load resistance are also investigated

(a) (b)

Figure 5. (a) Experimental setup used for validation of the analyticalmodel and the (b) bimorph with a tip mass attachment analyzed inthe experiment.

and predicted by using the analytical relations. Since thefundamental vibration mode of the harvester has the highestpractical importance for energy harvesting, attention is givento this mode. Variation of the voltage, current and poweroutputs with load resistance are investigated for excitations atthe short circuit and open circuit resonance frequencies of thefundamental mode. Optimum resistive loads of the harvesterare identified for excitations at these frequencies.

7.1. Experimental setup for a bimorph cantilever with a tipmass

The experimental setup used for measuring the voltage-to-base acceleration and tip velocity-to-base acceleration FRFsof the harvester is shown in figure 5(a). The bimorphanalyzed in this experiment is displayed in figure 5(b) andit is manufactured by Piezo Systems, Inc. (T226-A4-503X).The same type of bimorph was recently used by duToit et al[33] for the verification of their Rayleigh–Ritz model. Here,we attach a tip mass to the cantilever to make the problemrelatively sophisticated in terms of modeling (figure 5(b)). Thebimorph consists of two oppositely poled PZT-5A piezoelectricelements bracketing a brass substructure layer. Therefore, thepiezoelectric elements are connected in series as schematicallygiven in figure 1(a). The geometric and material properties ofthe piezoceramic and substructure layers are given in table 1.Note that, in agreement with the formulation given in thispaper, the length described by L is the overhang length ofthe harvester, i.e., it is not the total free length (63.5 mm) ofthe bimorph as acquired from the manufacturer. In addition,permittivity at constant strain is given in table 1 in terms of thepermittivity of free space, ε0 = 8.854 pF m−1 [16].

The bimorph cantilever is excited from its base with asine sweep generated by an electromagnetic LDS shaker. Thebase acceleration of the harvester is measured by a low massaccelerometer (PCB U352C22) and the velocity response ofthe harvester at the free end is measured by a laser vibrometer(Polytec OFV303 laser head, OFV3001 vibrometer). Theexperimental voltage FRF (in V/g) and tip velocity FRF (in(m/s)/g) obtained for a resistive load of 1 k� are shown infigure 6(a). The coherence functions of these measurements

12


Figure 6. (a) Experimental voltage and tip velocity FRFs of the cantilever and (b) their coherence functions (for a resistive load of 1 k�).

Table 1. Geometric and material parameters of the bimorph cantilever used for the experimental validation.

Geometricparameters Piezo. Substructure Material parameters

Piezo(PZT-5A)

Substructure(brass)

Length, L (mm) 50.8 50.8 Mass density, ρ (kg m−3) 7800 9000Width, b (mm) 31.8 31.8 Young’s modulus, Y (GPa) 66 105Thickness, h (mm) 0.26 (each) 0.14 Piezo. constant, d31 (pm V−1) −190 —Tip mass, Mt (kg) 0.012 Permittivity, εS

33 (F m−1) 1500ε0 —

are given by figure 6(b). The coherence is considerably lowfor frequencies less than 30 Hz but it is good around the firstresonance frequency (which is approximately 45.6 Hz for a1 k� resistive load).

The single-mode analytical FRFs given by equations (68)and (71) are used in order to validate their accuracy inpredicting the experimental observations. Note that the baseis not rotating and therefore θ0 = 0 in equations (67) and (70).The fundamental vibration mode (seen around 45.6 Hz infigure 6(a)) is of practical interest and consequently r = 1 isused in equations (68) and (71). It is important to note that thelaser vibrometer measures the absolute velocity at the tip of thebimorph in the experiment. However, the tip displacement FRFgiven by equation (71) for x = L is the displacement of the tiprelative to the vibrating base, i.e., it is not relative to the fixedframe. Hence, by considering the absolute displacement givenby equation (37), equation (71) must be modified as follows tocompare it with the experimental tip velocity measurement:

βmodifieds (ω, L) =

dws(L ,t)dt

−ω2Y0ejωt=

ddt [Y0ejωt + ws

rel(L, t)]−ω2Y0ejωt

= 1

jω+ jωβs(ω, L). (79)

Thus, the absolute tip velocity FRF given by equation (79)is used in comparisons with the laser vibrometer measure-ments. Note that, instead of modifying the analytical FRFexpression given by equation (71), one could as well processthe experimental FRF. However, this option is not preferablebecause of the possibility of generating noise while post-processing the experimental data. It should also be added that,in the following, the FRFs given by equations (68) and (79) aremultiplied by the gravitational acceleration (g = 9.81 m s−2)to be in agreement with the experimental measurements (hencethe FRFs are given per base acceleration in g). Comparisonof the experimental measurements and model predictions aregiven next.

7.2. Validation of the single-mode expressions and coupledanalysis of the harvester

Since the performance of the harvester at resonance is themain concern, accurate identification of mechanical dampingratio is very important. It is a common practice to extract themodal mechanical damping ratio from the first experimentalmeasurement. The uncoupled (but mechanically damped)natural frequency of the harvester can be observed in theexperimental FRF by setting Rl → 0 (short circuit conditions),i.e., by using a very low resistive load (since, practically, nowire with zero electrical resistance exists). The measurementprovided for 1 k� resistive load is close to short circuitconditions for the given harvester and the fundamental naturalfrequency in short circuit conditions can be extracted fromthe experimental voltage or tip velocity FRF (figure 6(a)) as45.6 Hz. By using the numerical data of the bimorph harvestergiven in table 1, equation (15) predicts the first uncoupled (andmechanically undamped) natural frequency of the harvesteranalytically as 45.7 Hz. One can then identify the mechanicaldamping ratio of the first mode by employing the coupledsingle-mode relations as 2.7%. Hence, this approach allowsextracting modal mechanical damping in the presence of afinite resistive load (without forcing the system exactly to bein short circuit conditions). Indeed, if the electromechanicalmodel is self-consistent, one must be able to identify themechanical damping ratio for any value of load resistance.Furthermore, either the voltage FRF or the tip velocity FRFcan be used for identifying modal mechanical damping ratio,since the bimorph harvester itself is a transducer. In otherwords, theoretically, the coupled tip velocity information isincluded in the voltage output information of the harvester, andthe voltage and tip motion predictions for the same mechanicaldamping ratio must be in agreement based on the linearelectromechanical system assumption.

The mechanical damping ratio of the first vibration modeis identified (as ζ1 = 0.027) by using the voltage FRF

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Figure 7. Comparison of the model predictions and experimental measurements; (a) voltage FRF and (b) tip velocity FRF for 1 k�,(c) voltage FRF and (d) tip velocity FRF for 33 k�, (e) voltage FRF and (f) tip velocity FRF for 470 k�.

as shown in figure 7(a) (for 1 k� load resistance). Forthis identified damping ratio, the voltage FRF of the model(obtained from equation (68)) is in perfect agreement with theexperimental FRF as shown in figure 7(a). As discussed inthe previous paragraph, for the same damping ratio (2.7%), thetip velocity FRF obtained from the model should predict theexperimental tip velocity FRF accurately. The tip velocity FRFobtained from equation (79) is plotted with the laser vibrometermeasurement in figure 7(b). As can be seen from thisfigure, the agreement between the theoretical and experimentaltip velocity FRFs is very good, which clearly shows theconsistency of the electromechanical model proposed here. Ifthe resistive load is replaced by a resistive load of 33 k�,the experimental and analytical voltage and tip velocity FRFsgiven by figures 7(c) and (d) are obtained, respectively. Notethat the mechanical damping ratio is kept at 2.7% in themodel and the model predicts the coupled structural responsesuccessfully for this different resistive load (which is oneorder of magnitude larger than the previous one). The shift

in the resonance frequency for a 33 k� resistive load is notvery large. However, if the resistive load is increased to470 k� (figures 7(e) and (f)), the resonance frequency movesto 48.4 Hz, which is approximately 2.8 Hz higher than theresonance frequency for 1 k�. Note that the system is closeto open circuit conditions for the large resistive load of thelast case (470 k�). The variations in the fundamental moderesonance frequency with changing load resistance as wellas the amplitude-wise results in the FRFs are successfullypredicted by the analytical model (modal mechanical dampingratio is kept constant at 2.7% in all cases).

From the quantitative point of view, the maximum voltageoutput increases from 1.57 V/g (at 45.6 Hz) to 84 V/g (at48.4 Hz) as the resistive load increases from 1 to 470 k�. Notethat the former case (1 k�) is close to short circuit conditions(corresponding to the highest current output) whereas the lattercase (470 k�) is close to open circuit conditions (yieldingthe highest voltage output). Therefore, the short circuit andopen circuit resonance frequencies for the first mode of this

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Figure 8. Enlarged views of the (a) voltage FRF and the (b) tip velocity FRF for 8 different values of load resistance (model predictions andthe experimental measurements).

Figure 9. Enlarged views of the (a) current FRF for 8 different values of load resistance and the (b) power FRF 3 different values of loadresistance (model predictions and experimental measurements).

harvester are approximately 45.6 Hz and 48.4 Hz, respectively.The analytical model predicts these two frequencies as 45.7 Hzand 48.2 Hz, respectively.

The experimental measurements are repeated for 8different values of load resistance: 1, 6.7, 11.8, 22, 33, 47,100 and 470 k�. Each of the resistive loads results in adifferent voltage FRF and a tip velocity FRF. Figures 8(a)and (b), respectively, display enlarged views of the voltageoutput and tip velocity FRFs around the first vibration modefor these 8 different values of load resistance. The directionof increasing load resistance is depicted with an arrow and itis clear from figure 8(a) that the voltage across the resistiveload increases monotonically with increasing load resistance atevery excitation frequency. For the extreme values of the loadresistance, the frequency of maximum voltage output movesfrom the short circuit resonance frequency to the open circuitresonance frequency. For a moderate value of load resistance,the frequency of maximum voltage has a value in betweenthese two extreme frequencies (i.e., between 45.6 and 48.4 Hzin this case). The shift in the frequencies of maximum responseamplitude is also the case in the tip velocity FRF (figure 8(b)).However, the variation of tip velocity with load resistance isnot necessarily monotonic at every frequency. For excitationat 45.6 Hz, the tip motion is suppressed as the resistive load isincreased up to a certain value. It is very important to note thatthis suppression in the motion amplitude is more sophisticatedthan viscous damping. With increasing load resistance, themotion is attenuated at 45.6 Hz whereas it is amplified at48.4 Hz. Hence, if one focuses on the open circuit resonance

frequency (48.4 Hz), both the voltage output and vibrationamplitude at the tip increases with increasing load resistance.Therefore, modeling the effect of piezoelectric coupling in thebeam equation as viscous damping clearly fails in predictingthis phenomenon (in addition to the fact that it cannot predictthe frequency shift due to changing load resistance). Notethat, for 8 different resistive loads, the model predicts thefrequency response of the voltage output and tip velocity verysuccessfully.

The electric current FRF exhibits the opposite behaviorof the voltage FRF with changing load resistance as shownin figure 9(a) (obtained from I = V/Rl). Hence, theelectric current decreases monotonically with increasing loadresistance at every excitation frequency. Figure 9(b) displaysthe electrical power FRF for 3 different resistive loads5. Thetrend in the electrical power FRF with changing load resistanceis more interesting as it is the multiplication of two FRFs(voltage and current) with the opposite trends. As can beseen in figure 9(b), the electrical power FRFs of differentresistive loads intersect each other just like the tip velocityFRF (figure 8(b)). For a given excitation frequency, thereexists a certain value of load resistance that gives the maximumelectrical power. This value is called the optimum loadresistance and it can be observed more easily if the frequency

5 In order to avoid confusion with 8 intersecting curves, the electrical powerFRF is given for 3 resistive loads only. Note that the electrical power amplitudeis due to P = |V |2/Rl; i.e., it is the peak power. The average power canbe obtained from Pave = |Vrms|2/Rl, where Vrms = V/

√2 (thus, Pave =

|V |2/2Rl = P/2).

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Figure 10. Variations of the (a) peak voltage, (b) peak current and the (b) peak electrical power amplitudes with load resistance for excitationsat the short circuit and open circuit resonance frequencies of the first vibration mode.

of interest is kept constant and the power amplitude is plottedagainst load resistance (which is addressed next).

The short circuit and open circuit resonance frequenciesof the first mode are defined for the extreme cases of loadresistance (45.6 Hz as Rl → 0 and 48.4 Hz as Rl → ∞) andthese frequencies are of practical interest. The variations ofthe voltage output with changing load resistance for excitationsat these two frequencies are shown in figure 10(a). In bothcases, voltage increases monotonically with load resistance.The voltage output for excitation at the short circuit resonancefrequency is higher when the system is close to short circuitconditions and vice versa. There exists a certain resistiveload (83.4 k�), for which the voltage response has the sameamplitude (40.6 V/g) for excitations at both frequencies. Themaximum voltage amplitude in the limit Rl → ∞ is about54.5 V/g for excitation at 45.6 Hz and it is about 108.8 V/g forexcitation at 48.4 Hz. Figure 10(b) shows the variations of theelectric current with changing load resistance for excitationsat these two frequencies. The trend of the current amplitudewith changing load resistance is the opposite of that of thevoltage amplitude. That is, the current amplitude decreasesmonotonically with increasing load resistance. The currentoutput for excitation at the short circuit resonance frequencyis higher when the system is close to short circuit conditionsand vice versa. Again, for an 83.4 k� load resistance,both excitation frequencies yield the same current amplitude(0.49 mA/g). In the limit Rl → 0, the maximum currentamplitude is about 1.57 mA/g for excitation at 45.6 Hz andit is about 0.68 mA/g for excitation at 48.4 Hz.

The variation of the electrical power with changing loadresistance is given in figure 10(c) for the short circuit andopen circuit resonance frequency excitations. As mentioned

before, the variation of the electrical power with changingload resistance is not monotonic. These two cases (the shortcircuit and open circuit resonance frequency excitations) havedifferent optimum resistive loads which yield the maximumelectrical power. The optimum load resistance for excitation at45.6 Hz is about 35 k�, yielding a maximum electrical powerof about 23.9 mW/g2 whereas the optimum resistive load forexcitation at 48.4 Hz is 186 k�, yielding approximately thesame power output. As in the case of voltage and currentoutputs, the electrical power output for excitation at the shortcircuit resonance frequency is higher when the system isclose to short circuit conditions and vice versa. Moreover,for an 83.4 k� resistive load, the same electrical power(19.8 mW/g2) is obtained for excitations at both of thesefrequencies. The respective trends in the electrical outputsat the short circuit and open circuit resonance frequencies ofthe first mode are successfully predicted by the single-modeanalytical relations derived in this paper.

A useful practice to obtain some additional informationregarding the performance of the harvester device impliesdividing the electrical power by the volume and by the mass ofthe harvester. The overhang volume of the bimorph cantileveris 1.07 cm3 whereas the volume occupied by the tip massattachment is 2.45 cm3, yielding a total device volume of about3.52 cm3. The overhang mass of the bimorph is 8.6 g and thetip mass is 12 g. Thus, the total mass of the cantilever is about20.6 g. The electrical power versus load resistance graph givenby figure 10(c) can therefore be plotted in the form of powerdensity (power per device volume) and specific power (powerper device mass) graphs. The vertical axis of figure 10(c) mustbe divided by the device volume to obtain the power densitygraph and it must be divided by the device mass to obtainthe specific power graph. The variations of power density and

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Figure 11. Variations of the (a) power density (power per device volume) and the (b) specific power (power per device mass) amplitudes withload resistance for excitations at the short circuit and open circuit resonance frequencies of the first vibration mode.

specific power with load resistance are given by figures 11(a)and (b), respectively (for the short circuit and open circuitresonance excitations). For instance, for excitation at 45.6 Hz,the maximum power density is about 6.8 (mW/g2) cm−3

and the maximum specific power is about 1.15 (W/g2) kg−1

(for a 35 k� resistive load). It is very important to notethat the power density and the specific power concepts arenot complete dimensionless representations. For instance, thesame device volume can be occupied by the same amountof material (piezoceramic, substructure and tip mass) for adifferent aspect ratio of the beam, yielding a larger or smallerelectrical power with totally different natural frequencies. Yet,these representations have been found useful for comparison ofthe harvester devices in the literature.

8. Summary and conclusions

Piezoelectric energy harvesting has been investigated byseveral researchers for the last five years. Typically, acantilevered harvester beam with one or two piezoceramiclayers is located on a vibrating host structure and the harvesterbeam generates electrical power due to base excitation.Electromechanical modeling of cantilevered piezoelectricenergy harvesters under base excitation has been studiedmy many authors and the existing models include SDOFapproaches, approximate solutions in the sense of Rayleigh–Ritz discretization and analytical solution attempts with certainsimplifications. Recently, the authors have presented theclosed-form analytical solution for a unimorph cantilever basedon the Euler–Bernoulli beam assumptions. In this work, theanalytical solution is extended to bimorph configurations withseries and parallel connections of piezoceramic layers andexperimentally validated.

The base excitation acting on the bimorph cantileveris assumed to be translation in the transverse directionwith superimposed small rotation. For series and parallelconnections of the piezoceramic layers, the closed-formelectromechanical expressions are first obtained for the steadystate response to harmonic excitation at arbitrary frequencies.The resulting expressions are then reduced to single-modeexpressions by assuming modal excitation (i.e., excitation at orvery close to a particular natural frequency), which is the main

concern in vibration-based energy harvesting. The single-mode relations given here are easier to use compared to themulti-mode solutions and they are as accurate as the multi-mode solutions for excitations around a natural frequency ofinterest (which, in general, is the first natural frequency ofthe harvester). The electromechanical FRFs which relate thevoltage output and vibration response of the bimorph to thetranslational and rotational base acceleration components areextracted both for the multi-mode and single-mode solutions.

In order to validate the model proposed in this paper, anexperimental study is presented for a bimorph cantilever witha tip mass attachment. It is shown that the single-mode FRFsobtained from the analytical solution given here can predictthe voltage output and the vibration response FRFs of thebimorph very accurately. The base excitation experiments arerun for 8 different resistive loads and it is shown that theanalytical model can successfully predict the variation in thecoupled electrical and mechanical response of the cantileveredbimorph. The outputs of the harvester device (current, voltageand power) are analyzed extensively for the short circuit andopen circuit frequency excitations and the accuracy of thesingle-mode relations is observed in all cases. Since theyare based on the distributed parameter solution, the single-mode electromechanical FRFs proposed in this paper here cantake the place of the elementary SDOF solutions for modalexcitations. Moreover, the single-mode expressions given hereare not limited to the fundamental mode and they can be usedfor any vibration mode as they originate from the distributedparameter solution. The multi-mode closed-form expressionsgiven here can be used if the same harvester is to be excitedat different vibration modes or at its off-resonance frequenciesdue to multi-frequency or varying-frequency inputs.

Acknowledgments

The authors gratefully acknowledge the support of the AirForce Office of Scientific Research MURI under grant numberF 9550-06-1-0326 ‘Energy Harvesting and Storage Systemsfor Future Air Force Vehicles’ monitored by Dr B L Lee.Mr A Erturk acknowledges the support from the ‘LiviuLibrescu Memorial Scholarship’ awarded by the Departmentof Engineering Science and Mechanics at Virginia Tech. The

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authors also thank Mr P Tarazaga for his help in preparing theexperimental setup.

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