Modeling and experimental verification of proof mass effects on vibration energy harvester

performance

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Smart Mater. Struct. 19 (2010) 045023 (21pp) doi:10.1088/0964-1726/19/4/045023

Modeling and experimental verification ofproof mass effects on vibration energyharvester performanceMiso Kim, Mathias Hoegen, John Dugundji and Brian L Wardle

Department of Aeronautics and Astronautics, Massachusetts Institute of Technology,Cambridge, MA 02139, USA

Received 3 November 2009, in final form 24 January 2010Published 16 March 2010Online at stacks.iop.org/SMS/19/045023

AbstractAn electromechanically coupled model for a cantilevered piezoelectric energy harvester with aproof mass is presented. Proof masses are essential in microscale devices to move deviceresonances towards optimal frequency points for harvesting. Such devices with proof masseshave not been rigorously modeled previously; instead, lumped mass or concentrated pointmasses at arbitrary points on the beam have been used. Thus, this work focuses on the exactvibration analysis of cantilevered energy harvester devices including a tip proof mass. Themodel is based not only on a detailed modal analysis, but also on a thorough investigation ofdamping ratios that can significantly affect device performance. A model with multiple degreesof freedom is developed and then reduced to a single-mode model, yielding convenientclosed-form normalized predictions of device performance. In order to verify the analyticalmodel, experimental tests are undertaken on a macroscale, symmetric, bimorph, piezoelectricenergy harvester with proof masses of different geometries. The model accurately captures allaspects of the measured response, including the location of peak-power operating points atresonance and anti-resonance, and trends such as the dependence of the maximal powerharvested on the frequency. It is observed that even a small change in proof mass geometryresults in a substantial change of device performance due not only to the frequency shift, butalso to the effect on the strain distribution along the device length. Future work will include theoptimal design of devices for various applications, and quantification of the importance ofnonlinearities (structural and piezoelectric coupling) for device performance.

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

1. Introduction

In recent years, harvesting ambient energy from theenvironment has been a growing interest area and major focusof many research groups [1]. Many ambient power sourcessuch as thermal gradients, mechanical vibrations, fluid flow,solar, human-driven sources [2, 3] etc. have been activelyinvestigated in order to realize alternative power supplies.Ambient sources are potential candidates to replace existingpower sources such as batteries that have a limited energystorage capacity and lifetime for some applications [4]. Inparticular, mechanical vibration energy harvesting has drawnmuch attention as substantial advances have been achievedin integrated circuit technology, particularly in low powerdigital signal processors, reducing power requirements for

wireless sensor nodes [5, 6]. Energy harvesting from externalmechanical excitation is made through conversion of nearlyubiquitous, ambient mechanical vibration energy using one ofthree transduction mechanisms: electrostatic, electromagnetic,or piezoelectric effects [1, 7]. Although each transductionmechanism and corresponding application has advantages indifferent areas, energy conversion using piezoelectricity isregarded as one of the most promising technologies for MEMSdevices. Piezoelectric materials [8] produce an electricalcharge or voltage when subjected to a mechanical stressor strain, or vice versa. Vibrational energy is directlyconverted to voltage with no need for complex geometries oradditional components. This is in contrast with electrostaticdevices where an input voltage is required. Perhaps mostimportantly, piezoelectric energy harvesters can generate high

0964-1726/10/045023+21$30.00 © 2010 IOP Publishing Ltd Printed in the UK & the USA1

Smart Mater. Struct. 19 (2010) 045023 M Kim et al

output voltages and enhanced efficiency compared to otherenergy harvesting schemes [7].

In order to harvest mechanical vibration energy utilizingthe piezoelectric effect, researchers have developed devicesfor various applications including windmills, shoe inserts,implantable devices, etc [1]. Especially, for applications inmicrosystems, several studies have focused on developingMEMS piezoelectric energy harvesters using establishedpiezoelectric film processing [9–13]. Along with researchon fabrication of devices, researchers have also putconsiderable amount of effort in developing analytical modelsfor piezoelectric energy harvesters to study the dynamiccharacteristics of these structures. Several models have beenproposed and applied not only to predict and analyze thedevices but also to optimize the design for future applications.Equivalent electrical circuit modeling has been used to describethe mechanical components of the mechanical elements torepresent the system purely in the electrical domain [14–18].Alternatively, Euler–Bernoulli beam theory [19] has also beenapplied to study the dynamics of piezoelectric energy harvesterin combination with either force equilibrium analysis [20]or via energy methods [21]. A coupled electromechanicalmodel, based on a structural modal analysis for a base-excitedcantilever, was developed by du Toit et al [24, 25]. Cantileverbeam and plate configurations were chosen not only because itis geometrically compatible with MEMS fabrication processesbut also because such a compliant structure can produce highstrain, and thus more power generation, in comparison withother structural configurations. Depending on the number ofpiezoelectric element layers, the structure can be categorizedas either a unimorph or bimorph configuration. Two modes oftransduction can be used according to the direction of electricalfield and applied strain: {3-1} mode of operation and {3-3}mode of operation.

Useful closed-form analytic expressions for key deviceperformance characteristics were previously presented usingthe energy method approach focusing on a single vibrationalmode [22, 23], from which both electrical and mechanicalperformance could be predicted and optimized across electricalloading conditions and frequencies. This power-optimizedelectromechanically coupled model was experimentally veri-fied using a macroscopic, symmetric bimorph, piezoelectric{3-1} device. The verified model, in a convenient normalizedform that allows clear interpretation of the device performance,may also be used for proper experimental design as in thecurrent work. Other investigations of similar piezoelectricenergy harvester configurations are given by [26–28], withsimilar results but omitting the normalization and many keytrends (e.g. resonance and anti-resonance operating points ofinterest).

In the current work, a rigorous treatment of a proof massis presented in detail, including experimental verification. InMEMS applications, as the device dimensions scale down, thenatural frequencies of these devices can approach the GHzrange. In order to use these devices for practical applications,it is desirable to adjust the natural frequencies to lower levelswhere more significant energy is found in typical spectra [22].A beam end mass, so called a proof mass, serves to decrease

Figure 1. Schematic of base-excited cantilevered beam/plate with aproof mass at the tip.

the resonance frequency of a device and also to raise theaverage level of inner stress and strain along the beam lengthto increase power generated. While it is very common to attachor integrate a proof mass at the end of the beam/plate to obtainthe target frequency of interest [10–12], a rigorous analyticaltreatment of a cantilevered harvester with a proof mass hasnot been presented. An exact treatment of the proof mass isconsidered here resulting in modification of calculated terms inthe prior normalized governing equations [25]. It is shown thatwhile the addition of the proof mass modifies numerous termsin the governing equations, the normalized results still hold.Further, a detailed set of experiments is performed to verifythe model by varying proof mass geometry. Additionally,we present a generalized treatment for the determination ofmechanical damping ratios based on simple measurement ofactual tip displacements and base displacements originated indu Toit et al [22–25]. Modeling is an indispensable elementin predicting and analyzing the characteristics of piezoelectricmechanical vibrations energy harvesters. Therefore, the exactanalysis presented here for piezoelectric mechanical vibrationenergy harvesters with a proof mass is an essential step inrealizing MEMS devices for future applications.

2. Modeling of cantilevered piezoelectric vibrationenergy harvesters with a proof mass

In this section, a coupled electromechanical modal model isbriefly reviewed before a detailed derivation to incorporate theproof mass is undertaken. The derivation is based on Euler–Bernoulli theory and an energy method approach. Then, keymechanical and electrical device characteristics are extracted.While we begin with treating the general multi-degree-of-freedom system, we focus on the closed-form analytic solutionin a single mode model that has been shown to be adequateto predict device performance in past work. The model isimplemented for the case of various energy harvesters builtand tested for model verification. Each energy harvesteris composed of piezoelectric element, structural layer, andelectrodes and the device is wired to the appropriate electricalcircuit to extract the electrical properties such as voltage andpower. The input power source comes from base-excitedmechanical vibrations (see figure 1).

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

2.1. Modeling cantilever beam and plates with piezoelectricelements

There are two methods to obtain the model for a cantileverbeam/plate with piezoelectric elements: an energy methodapproach and a force equilibrium analysis [20]. Our study isbased on the energy method approach, following the energyformulation for actuators by Hagood et al [21] and developedby du Toit et al [25] for energy harvesting. According to theenergy conservation law, an electromechanical system can bemodeled in terms of the sum of kinetic energy (Tk), internalpotential energy (U ), electrical energy (We) and external work(W ) as follows:

∫ t2

t1[δ(Tk − U + We)+ δW ] dt = 0. (1)

Individual energy terms and external work term are defined as:

Tk = 12

∫Vs

ρsutu dVs + 12

∫Vp

ρputu dVp (2)

U = 12

∫Vs

StT dVs + 12

∫Vp

StT dVp (3)

We = 12

∫Vp

EtD dVp (4)

δW =n f∑

k=1

δukfk(t)+nq∑j=1

δϕ j q j . (5)

Vp and Vs are volumes of piezoelectric element sectionand of the structural (inactive) section of the beam/plate,respectively. Likewise, ρp and ρs denote densities of eachsection. Dots indicate the time derivatives and u(x, t) is therelative displacement matrix, and u represents the velocity.It should be noted that the relative displacement of thestructure, u(x, t) can be written as the sum of nr individualmodes shapes, ψri (x), multiplied by a generalized mechanicalcoordinate, ri (t), using the Raleigh–Ritz approach. The modeshape is a function only of the axial position because only thetransverse displacement (xt direction) is considered in bendingof a beam/plate. Thus, u(x, t) can be expressed as w(xa, t),which is the beam displacement relative to the base of thebeam/plate, where xa is the axial beam/plate coordinate. Thebase excitations follow similarly,wB and thuswB = wB(xa, t).The relative displacement above is given in equation (6):

u(x, t) = w(xa, t) =nr∑

i=1

ψri (xa)ri (t) = ψ r (xa)r(t) (6)

whereψr (xa) represents the row matrix, [ψr1(xa),ψr2(xa), . . .]of mode shapes. Equations (3) and (4) represent the internalpotential and electrical energies in terms of S, T, E, and Dmatrices. The S, T, E, and D matrices are defined as appliedstrain, developed stress, applied electric field, and developedelectric displacement, respectively and the superscript, t, hereindicates the transpose of the matrix. According to 3D linearelastic (small-signal) constitutive relations, these matrices arerelated through electrical parameters such as the permittivity

of the piezoelectric element, ε, the piezoelectric constantrelating charge density and strain, e, and the mechanicalparameter, cE, the stiffness matrix. These physical parametersare obtained differently depending on whether the structure isa beam or a plate (see discussion in [22, 24]). Equation (7)is the representative expression for three-dimensional linearelastic constitutive relations where a range of small-signalpiezoelectric motion is assumed.

{TD

}=

[cE −et

e εS

]{SE

}. (7)

Superscripts E, S indicate parameters at constant electric fieldand constant strain. The three-dimensional linear elasticconstitutive relations as in equation (7) can be easily simplifiedfor {3-1} and {3-3} modes of operation where equation (7) issignificantly simplified [22, 23]. Strain, S, in particular, canalso be expressed with respect to mechanical mode shapes.The Euler–Bernoulli beam theory allows the axial strain inthe beam to be written in terms of the beam neutral axisdisplacement and the distance from the neutral axis (xt:transverse direction, see figure 1), as given by equation (8).Primes represent the spatial derivatives throughout this paper.

S(x, t) = −xt∂2w(xa, t)

∂x2a

= −xtψ′′r r(t). (8)

Lastly, in order to define the external work term in equation (5),nf discretely applied external point forces, fk(t) at positionsxk , and nq charges, q j , extracted at discrete electrodes withpositions x j are introduced. In this same equation for theexternal work term, the quantity ϕ j = ϕ(x j , t) is the scalarelectrical potential for each of the nq electrode pairs. For {3-1} mode devices as used here (see figure 2), there is only oneelectrode pair on either surface of the piezoelectric element,while for a {3-3} mode test device, there may be a largenumber of electrode pairs distributed over one surface of thepiezoelectric layer. This scalar electrical potential term, ϕ j =ϕ(x j , t) can be expressed via a potential distribution, ψv j (x),and the generalized electrical voltage coordinate, v j (t) as inequation (9):

ϕ(x, t) =nq∑j=1

ψv j (x)v j(t) = ψv(x)v(t). (9)

It is important to differentiate electrical mode shape vector,ψv(x) from the mechanical mode shape vector, ψ r (x). Theform of electrical mode shapes varies according to the specificsof test-devices, and will be detailed in later sections.

When we substitute equations from (6) to (9) intoequations (2)–(5), the energy and external work expressionsin equations (2)–(5) can be rewritten in terms of mechanicalor electrical mode shapes and material parameters. Theserewritten equations (2)–(5) are then inserted back into theenergy conservation equation (1). Finally, rearrangement ofequation (1) allows us to obtain two governing equations ofmotion as below.

Mr + Cr + Kr − Θv = −BfwB (10)

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

Figure 2. Illustration of symmetric, bimorph energy harvester with a tip proof mass. The active elements are electrically connected in series.

Θtr + CPv + q = 0. (11)

As defined in equation (6), r denotes the generalized relativedisplacement vector. As shown in equation (6), in order toconvert it to actual displacements, it is necessary to multiply itby the mechanical mode shapes. Exact modal analysis will befollowed in section 2.2. v is the developed voltage across thepiezoelectric element and q is the charge. Base acceleration,wB, is the input to the cantilevered system (see figure 1).Several effective terms, including the mass (M), the stiffness(K), coupling (Θ), and capacitive matrices (CP), are definedbelow. The forcing vector, Bf, accounts for inertial loading onthe beam/plate structure due to the base excitations.

M =∫

Vs

ψ trρsψr dVs +

∫Vp

ψ trρpψ r dVp (12)

K =∫

Vs

(−xtψ′′r )

tcs(−xtψ′′r ) dVs+

∫Vp

(−xtψ′′r )

tcE(−xtψ′′r ) dVp

(13)

Θ =∫

Vp

(−xtψ′′r )

tet(−∇ψv) dVp (14)

CP =∫

Vp

(−∇ψv)tεS(−∇ψv) dVp (15)

Bf =∫ L

0m(xa)ψ

tr dxa. (16)

Mechanical mode shapes, electrical mode shapes, andtheir derivatives mainly comprise these terms. Details onconventions of materials properties used here can be foundelsewhere [8, 23, 25]. It should be noted that mechanicaldamping is included by adding a viscous damping term, C.As it is assumed that the damping has little dependence onthe device natural frequency, damping is typically measuredat the device natural frequency, which will be covered indetail in a later section. In the general multi-degree-of-freedom (MDOF) governing equations obtained above, themechanical domain represented by the actuation equation (10)is electromechanically coupled to the electrical domainexpressed by the sensing equation (11) via the coupling term(Θ). As the electromechanical coupling stems from the

piezoelectric element, the coupling term (Θ) is directly relatedto the piezoelectric constant (e) as shown in equation (14).Evaluation of the above coefficients from equations (12) to (16)for practical applications will be presented in later sections,using a single beam mode (ψr1).

2.2. Modal analysis of cantilevers with a proof mass

Effective terms in the governing equations are generallyexpressed in terms of mechanical mode shapes and thus, it isenough to say that the extent of prediction capability of themodel depends significantly on the mode shapes. Altering themode shape via the addition of a proof mass will stronglyaffect all the effective constants in equations (10) and (11).While modal analysis and natural frequency calculations offree-clamped cantilevered structures appear in the literature(e.g. [19]), vibration analysis of simple beams with the additionof a proof mass is less common. Many researchers apply apoint mass assumption or lumped mass assumption in theirstudy on modal analysis and natural frequencies. In thissection, it will be shown how much the material and geometricproperties of a proof mass and the location where it is placed inthe system affect the modal analysis and therefore, the analysisof the entire system. The proof mass is considered rigid andcontains rotational terms (due to offset center of gravity andattachment point, see figure 1), instead of a lumped mass orconcentrated point mass at a certain point.

First of all, it is assumed that the stiffness of a proof massis much higher than that of beam/plate itself so that proofmass contributes to the entire system with regard to the massincrease, and makes the beam/plate efficiently rigid in the outerportion from L to L0. (Future work will deal with non-rigidmass case when a proof mass also makes contributions to thebending stiffness of the system.) In order to treat more generalcases, it is not assumed that the proof mass center of gravitycorresponds to the point of loading on the beam/plate. Also, forsimplicity, the proof mass is assumed to be uniform in the axialdirection with mass per length, equal to m0, like the beam/platewhere the cross-section is considered to be uniform with massper length, equal to m (see figure 1). It should be emphasizedthat in this schematic, the part of the beam/plate under the areaof proof mass is considered part of the total proof mass for this

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

beam/plate system for calculation ease. Now we begin withthe introduction of mass of overhang, M0, static moment at thejunction x = L, S0, and moment of inertia at the junction, I0,to account for the properties of the total proof mass. In thisparticular illustration, we can express these terms as follows:

M0 = m0 L0 + mL0 (17)

S0 = M0L0

2(18)

I0 = m0 L0

3(L2

0 + h20)+

mL0

3(L2

0 + h2) (19)

where subscript 0 indicates the part of a proof mass, andh, L, b and ρ denote height, length, width and density ofthe beam/plate and the proof mass, respectively in order. Itis significant to note that the moment of inertia (I0) at thejunction (xa = L) between the beam/plate and the proof massincorporates the rotational inertia of the structure.

As adopted from [29, 30], the governing equations interms of mechanical displacement or mode shape can bedetermined, using Euler–Bernoulli beam theory, as given byequation (20). The N th mode of the mechanical modeshape is represented by ψr N (xa), which can be expressed viaequation (21) where constants (c, d , e, and f ) will be solvedusing the boundary conditions.

E Iψ IVr N − mω2

Nψr N = 0 (20)

ψr N = c sinhλN xa + d coshλN xa + e sinλN xa + f cos λN xa.

(21)E I is the effective bending stiffness, which is obtainedconsidering the neutral axis and the properties of the beammulti-layers, while ωN represents the N th mode resonancefrequencies. It is convenient to define the parameter, λ4

N =mω2

N

E I. Boundary conditions at the fixed end and at the junction

point where the beam/plate and the proof mass are connectedare given by:

At xa = 0,ψr N = 0 (22)

ψ ′r N = 0 (23)

At xa = L,

E Iψ ′′r N = ω2

N I0ψ′r N + ω2

N S0ψr N (24)

E Iψ ′′′r N = −ω2

N M0ψr N − ω2N S0ψ

′r N (25)

where ( )′ = ∂/∂xa. Equations (22) and (23) simply represent aclamped cantilever, whereas equation (24) represents bendingmoment and equation (25) represents shear force at thejunction between the beam/plate and the proof mass. Theseboundary conditions give four equations which can be reducedto the 2 × 2 matrix equation (26) along with the expressions ofmatrix elements, equations (27)–(30).[

A11 A12

A21 A22

] [cd

]= 0 (26)

A11 = (sinhλN + sin λN )+ λ3N I0(− coshλN + cosλN )

+ λ2N S0(− sinhλN + sinλN ) (27)

A12 = (coshλN + cos λN )+ λ3N I0(− sinhλN − sinλN )

+ λ2N S0(− coshλN + cosλN ) (28)

A21 = (coshλN + cos λN )+ λN M0(sinhλN − sinλN )

+ λ2N S0(coshλN − cos λN ) (29)

A22 = (sinhλN − sin λN )+ λN M0(cosh λN − cos λN )

+ λ2N S0(sinhλN + sin λN ). (30)

For convenience, all terms above are nondimensionalized suchthat λN = λN L, M0 = M0

mL , S0 = S0mL2 , and I0 = I0

mL3 .

We solve for λN which makes the determinant | A11 A12A21 A22

| =0 and thus obtain successive values of λN , from whichthe resonance frequencies associated with each N th mode,

ωN can be calculated through the relation, λ4N = λ4

N L4.The first resonance frequency, ω1, for example, is givenby equations (31) and (32), in units of (rad s−1) and (Hz),respectively:

ω1 = (λ1)2

√E I

mL4(rad s−1) (31)

f1 = 1

2πω1 (Hz). (32)

It should be noted that ω1 is also related to mass and stiffnessthrough ω1 = √

K/M . Then, the general bending mode shapeof a clamped-free (simple cantilever) beam/plate with a proofmass at the tip will be given as follows:

ψr N = d[(coshλN x − cosλN x)− A12

A11(sinh λN x − sinλN x)].

(33)This is a rewritten form of equation (21) only in terms of asingle arbitrary scaling constant, (here, d is used), and it isnoted that λN xa = λN

xaL . Once the parameter, λN or λN is

known from the zero determinant, both mode shape and theresonance frequency for the N th mode are readily obtainable.For practical use, we normalize the mode shape toψr N (L) = 2at the junction, which gives d = 2/[(coshλN L − · · ·]. Theactual beam/plate tip deflection at x = L + L0, is then givenby wtip = [2 + L0ψ

′r N (L)]r , rather than simply wtip = 2r as

used in the no proof mass case.Addition of a proof mass will cause the effective mass of

the structure, previously indicated as equation (12), to have adifferent form as shown in equation (34), which is obtainedfrom the Lagrange equations of motion.

M =∫

Vs

ψ trρsψ r dVs +

∫Vp

ψ trρpψ r dVp + M0(ψ r(L))

tψ r(L)

+ 2S0(ψ r(L))tψ ′

r(L)+ I0(ψ′r(L))

tψ ′r(L). (34)

The last three terms in equation (34) incorporate the propertiesof the proof mass. Although the first two terms are presentas in equation (12) when a proof mass is not considered, theresulting values of these terms will be quite different whena proof mass is added to the cantilevered structure. This isbecause the mode shapes vary due to material and geometricproperties of the proof mass and thus affect the first two termsin equation (34), which are expressed in terms of these mode

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

shapes. Likewise, as other terms in the governing equations ofmotions in section 2.1, such as effective stiffness (K), coupling(Θ), and capacitive matrices (CP), depend on the mode shapes,the variation of the mode shape will alternate the values forthese effective terms when there is a proof mass. Previously,the forcing function, Bf, has been defined to account for theinertial loading of the device due to base excitation. Since notonly a beam/plate but also a proof mass at the tip contributesto the inertial loading of the device, the forcing vector mustbe modified so that the displacement and the rotation of thetip mass are taken into account. The last two additional termsmodify the forcing vector as demonstrated in equation (35).

Bf =∫ L

0m(ψ r(xa))

t dxa + M0(ψ r(L))t + S0(ψ

′r(L))

t. (35)

The matrices M, K, Θ, and CP in equations (10)–(15),and (34) represent square matrices, while Bf in equations (16)and (35) represents a column matrix. Additionally, the massand stiffness matrices M and K are diagonal matrices becauseof the orthogonality conditions that exist between any twomodes, ψri and ψr j . Since each mode satisfies the differentialequation (24) and the boundary conditions equations (22)–(25),one can readily show the orthogonality conditions to result in

Mi j = δi j Mii (36)

Ki j = δi j Miiω2i (37)

where δi j represents the Kronecker delta. The fact thatthe mass and stiffness matrices, M and K, are diagonal isimportant in that it allows each vibration mode ri in thegoverning equation (10) to be uncoupled structurally fromone another (orthogonal vibration modes) in the governingsystem of equations (10) and (11). This allows for easiercomputation of the governing equations in the multi-degree-of-freedom (MDOF) treatment of the system.

2.3. Electromechanical model for single mode and poweroptimization

Expressions for single beam/plate mode can be obtained ifwe approximate the infinite degree-of-freedom mechanicalsystem as a single-degree-of-freedom (SDOF) system, wherethe multi-degree-of-freedom (MDOF) governing equations ofmotion are reduced to scalar forms. In order to applythe coupled model presented, it is more practical to havescalar equations, which allow simple optimization in termsof maximum power extraction. Single mode solutions havebeen shown to be in excellent agreement with the resultsfor devices without a proof mass. In single beam mode forbase-excited energy harvester structures, scalar expressions inequations (38) and (39) replace previously presented multi-degree-of-freedom (MDOF) equations (10) and (11).

Mr + Cr + Kr − θv = −BfwB (38)

θ r + CPv + 1

Rlv = 0. (39)

Note that sensing equation (39) is modified from equation (11)using Ohm’s law, v = Rl

dqdt , with a purely resistive electrical

load, Rl. Each coefficient in the single mode governingequations above can be expressed as below:

M =∫ L

0mψ2

r dxa + M0(ψr (L))2

+ 2S0ψr (L)ψ′r (L)+ I0(ψ

′r (L))

2 (40)

K = Mω21 (41)

θ =∫

−xtψ′′r e31(−∇ψv) dV (42)

Cp =∫(−∇ψv)εS

33(−∇ψv) dV (43)

Bf =∫ L

0mψr dxa + M0ψr (L)+ S0ψ

′r (L). (44)

In the above, ψr and ψv will from now on refer to the firstmode, ψr1 and ψv1. It should be noted that all terms presentedabove are applicable not only to the system where a proof massis attached at the end but also to no proof mass case. If weset all the parameters related to a proof mass as zero, then theexpressions coincide with the experimentally validated modelpresented by du Toit et al [25]. As electrical mode shape,ψv , highly depends on device specifics including electricalconnections between piezoelectric layers and mode operations,coupling term, θ , and capacitance term, Cp, will also varydepending on test device setups. Geometric configurationsof energy harvesters, such as unimorph or bimorph structuresas well as inter-element connection also affect the effectiverepresentations for coupling and capacitive terms. Therefore,suitable equations for coupling and capacitance term thatcorrespond to our test device specifics will be given insection 2.4.

From the governing equations (38) and (39), convenientclosed-form solutions of the amplitude for relative displace-ment, voltage developed and power extracted can be obtained.The magnitude of the power is calculated above as Pout =v2/Rl. These solutions for the mechanical and electricalperformance of the energy harvesters are:∣∣∣∣∣

rBfwB

∣∣∣∣∣ = 1

K

√1 + (α )2

[(1 − 2)− 2ςmα 2]2 + [({1 + κ2} − 2)α + 2ςm ]2

(45)∣∣∣∣∣v

BfwB

∣∣∣∣∣ = 1

|θ |ακ2

[(1 − 2)− 2ςmα 2]2 + [({1 + κ2} − 2)α + 2ςm ]2

(46)∣∣∣∣∣Pout

(BfwB)2

∣∣∣∣∣ = ω1

K

ακ2 2

[(1 − 2)−2ςmα 2]2+[({1 + κ2} − 2)α + 2ςm ]2.

(47)

In the above expressions, dimensionless factors are defined andused such that α = ω1 RlCp for dimensionless time constant,

κ2 = θ 2

K Cpfor system coupling, and = ω

ω1for dimensionless

frequency ratio. Also, mechanical damping ratio, ζm appearsand it is related to the damping constant, C via ςm =

C2Mω1

. Mechanical response is calculated using the generalizedmechanical displacement, r , which should be multiplied bynormalized mode shape to yield relative displacements, w, thatis, w = ψr (x)r . The magnitude of voltage (v) and power(Pout) are evaluated in a closed-form solutions as well, as givenin equations (46) and (47). Note that all these equations forsystem responses are nondimensionalized with an inertia force

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

represented either by BfwB or (BfwB)2. It is quite beneficial to

be able to use these closed-forms that are properly normalizedas we can avoid complications generated from the calculationsof complex numbers [28] in order to obtain the results. Also,the way the denominator appears in all three expressions is alsonoteworthy. As presented previously by du Toit et al [22–25],two optimal frequency ratios for equal maximum powergeneration are gained at resonance and anti-resonance, whichcorrespond to r = 1 and ar = √

1 + κ2, respectively. Thenatural frequency will correspond to either the resonance oranti-resonance frequency, depending on the electrical loadingfor a piezoelectric structure. These two frequency ratiosare obtained from the analysis of the system at short-circuitcondition (R → 0) and open-circuit condition (R → ∞).As natural frequencies are generally defined by the ratio ofstiffness to mass of the system as in ω = √

Keff/M , the originof these two natural frequencies for an electromechanicallycoupled system can be explained from the perspective ofstiffness, Keff. Two different effective stiffnesses, Keff, areobtained for short-circuit condition and open-circuit condition,which are K and K (1 + κ2), respectively. These stiffnessesstem directly from the piezoelectric constitutive relations thatalways specify the electrical or stress boundary condition atwhich the piezoelectric constant is measured [31]. Sincethe terms inside the denominator of equations (45)–(47) aresuccinctly arranged focusing on these two optimal frequencyratio expressions, it is readily possible to analyze the systemresponses at optimal points for maximum power extractionfrom the equations (45)–(47). In order to derive the maximumpower value, it is necessary to optimize the power equation (47)with respect to load resistance (Rl), which yields an optimalelectrical load in terms of dimensionless time constant, αopt.

α2opt = 1

2

(1 − 2)2 + (2ζm )2

([1 + κ2] − 2)2 + (2ζm )2. (48)

Substitution of suitable frequency ratios at both resonance( r = 1) and anti-resonance ( ar = √

1 + κ2) intoequation (48) yields two optimal electrical loadings, αopt,r atresonance and αopt,ar at anti-resonance:

α2opt,r = 4ζ 2

m

4ζ 2m + κ4

(49)

α2opt,ar = κ4 + 4ζ 2

m(1 + κ2)

4ζ 2m(1 + κ2)2

. (50)

Note that both optimal electrical loading conditions aredependent on mechanical damping as well as system coupling,and are roughly reciprocals of one another in α2

opt. Oncethese values are substituted back into the power equation (47),optimal power expression at resonance and anti-resonance aregained. In general, mechanical damping ratio, ζm, is at leastan order of magnitude smaller than square of system coupling,κ2, which allows us to approximate equations (49) and (50)to give simpler expressions for power as well. Using theapproximation, 2ζm/κ

2 � 1, equations (49) and (50) may besimplified to produce optimized power both at resonance and

anti-resonance as in the following.∣∣∣∣ Pout

(BfwB)2

∣∣∣∣opt,r

= ω1

K

1

8ζm(51)

∣∣∣∣ Pout

(BfwB)2

∣∣∣∣opt,ar

= ω1

K

12ζmκ4

4κ4 + 4ζ 2m(1 + κ2)

≈ ω1

K

1

8ζm. (52)

As pointed out previously by du Toit et al [22], the valuesof optimal power generated at resonance and anti-resonancehave the equal magnitude within the given assumptions. It hasbeen shown that for extremely high values of damping (perhapswhen the harvester is used as a damper, not seen in typicalmacroscale harvesting devices, and certainly not in MEMSdevices that have high quality factors), that the approximationabove does not hold [28].

2.4. Model implementation of experimental device

A symmetric, bimorph cantilevered PZT bender is chosenand tested with various sizes of proof mass as well aswithout a proof mass. The harvester is a cantileveredplate consisting of two piezoelectric layers (bimorph), onestructural layer, and electrodes (see figure 2). The testdevice is operated in {3-1} mode of operation with seriesconnection between active elements, which requires twopiezoelectric elements to be oppositely poled. In order toimplement the electromechanically coupled model based onexact modal analysis that can accommodate the specific testdevice configurations and interconnection of the piezoelectricelements, it is necessary to evaluate the relevant coefficients.In this section, we obtain appropriate scalar equations for theexperimental device.

Since there are two piezoelectric elements, each layer canbe denoted subscripts 1 and 2, to treat the two active layers inthe bimorph configuration. Considering these two layers forthe bimorph configuration, we write equations (53) and (54)as:

Mr + Cr + Kr − (θ1v1 + θ2v2) = −BfwB (53){θ1

θ2

}r +

[Cp1 00 Cp2

]{v1

v2

}= −

{q1

q2

}. (54)

Equation (54) are summed to obtain a form similar to the scalarequation (39), and effective constants derived:

(θ1 + θ2)r + (Cp1v1 + Cp2v2) = −(q1 + q2). (55)

For the symmetric bimorph configuration, θ1 = θ2, Cp1 = Cp2,v1 = v2, and q1 = q2. Next, the coefficients for one of thepiezoelectric elements (here 1 is chosen) need to be consideredso that they can be applied in the governing equations. Forbimorph cantilevered energy harvesters, there are two ways tointerconnect the piezoelectric elements: the parallel connectionand series connections. As the test device in this work isconnected in series, the poling directions of each piezoelectriclayer should be opposite. Please refer to [8, 23, 25] for detailedinformation of the effect of poling direction and correspondingelectrical connection. In series connection, as illustrated infigure 2, the device is connected to the electrical load by

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

shorting the two center electrodes (e2 and e3), and connectingthe top (e1) and bottom electrodes (e4) across the electricalload. Based on this, relations of voltage and charge areachieved such that v1 + v2 = v, and q1 = q2, giving:

Mr + Cr + Kr − θ1v = −BfwB (56)

θ1r + 1

2Cp1v + 1

Rlv = 0. (57)

Based on the comparison between the above expressions (56)and (57) and the governing equations (38) and (39) for generalsingle mode, it is recognized that the coupling term, θ , and thecapacitance term, Cp in general expression can be rewrittenusing coupling and capacitance values for one piezoelectriclayer through θ = θ1, and Cp = 1

2 Cp1 for the energy harvesterwith bimorph configuration that is electrically connected inseries. This result is intuitive in that two piezoelectric layers areregarded as simple capacitors and that the effective capacitanceof two identical simple capacitors wired in series is representedwith half of capacitance for one capacitor. As a next step, it isnecessary to examine the appropriate expression for electricalpotential that corresponds to symmetric, bimorph configurationin {3-1} mode of operation in order to evaluate coupling andcapacitance more specifically. In this work, electric potentialdistribution is taken to give a constant electric field throughthe thickness of the piezoelectric element. The potential variesfrom 0 at the top electrode to +1 of at the bottom electrode infigure 2. The electric potential for bimorph structure in {3-1}mode operation is then expressed employing the thicknesses ofpiezoelectric layer (tp) and structural layer (ts) as given by:

ψv =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(tp + ts2 )− xt

tpfor xt > 0

− ts2 − xt

tpfor xt < 0.

(58)

If the device is of unimorph configuration or in {3-3} modeof operation, electric potential should be differently expressedaccording to each case. Now it is possible to determine thefinal expression for coupling term, θ , and the capacitanceterm, Cp, by substituting electric potential expression shownin equation (58) into equations (42) and (43). The resultingexpressions are:

θ = θ1 = e31

(tp + ts

2

)bψ ′

r (L) (59)

Cp = 1

2Cp1 = 1

2

b(L + L0)

tpεS

33. (60)

Concerning with the calculation of the capacitance, the entirebeam/plate length, L + L0, is employed because of the testdevice structure. Thus, the model is implemented for the testdevice, that is, a symmetric, bimorph configuration in {3-1}mode of operation, connected electrically in series. Based onthis, mechanical and electrical performances of the system aresimulated and compared with the experimental results, as willbe presented in later sections.

2.5. Damping ratio evaluation with a proof mass

When the governing equations of motion are derived,mechanical damping is incorporated through addition of aviscous damping term, C, which is related to mechanicaldamping ratio, ζm, via C = 2ςmω1 M in the case of a singlemode. Mechanical damping is assumed to encompass allnon-electrical damping that influences the system response.Accurate characterization of mechanical damping is essentialfor device performance prediction. As an alternative to the log-decrement method (described for energy harvesters in detailin [23]), another measurement scheme can be developed toobtain ζm through convenient dynamic tests of the harvester.The relative displacement is expressed as a function ofdimensionless time constant, α, in equation (45). At short-circuit (α = 0) where electrical damping is zero, and atresonance ( = 1), equation (45) can be rewritten:

∣∣∣∣ r

BfwB

∣∣∣∣ = 1

K

1

2ζm. (61)

With an input harmonic base excitation given by wB =−ω2

1wB, where wB is amplitude of base excitation and ω1

denotes the first resonant frequency of the vibration. Inaddition to the base acceleration, wtip = ψtipr and K = Mω2

1can be substituted into equation (61) to derive an expressionfor the mechanical damping ratio, ζm, given easily measurabledisplacement amplitudes:

ζm = Bf

M

ψr,tip

2

∣∣∣∣ wB

wtip

∣∣∣∣ = Bf

M

ψr (L + L0)

2

∣∣∣∣ wB

wtip

∣∣∣∣. (62)

Absolute tip displacement, wtip, and absolute base displace-ment,wB, should be experimentally measured at resonance andshort-circuit conditions by aligning the input frequency withthe resonant frequency of the device and letting the electricalresistance, Rl be zero. Other values such as forcing function,Bf, and effective mass, M , in the equation can be obtainedfrom model implementation on specific devices. Equation (62)is advantageous in that it can be applied to energy harvestersystem not only when proof mass is added but also when thereis no proof mass. When a proof mass is attached at the end, themechanical mode shape at the tip, ψr,tip is equal to ψr (L + L0)

instead of simply ψr (L) = 2, as mentioned earlier, followingequation (33). For the energy harvester system without a proofmass, where L0 is zero, the mechanical mode shape at the tipreduces to 2 through ψr (L) = 2:

ζm ≈ Bf

M

∣∣∣∣ wB

wtip

∣∣∣∣. (63)

Equation (63) coincides with the form that was applied ina prior work by du Toit et al [25] for the case of noproof mass, where the experimental results agreed well withsimulated results. Note that in both cases, the mechanicaldamping has dependence on effective mass, M , and forcingfunction, Bf, that are represented in terms of mechanicalmode shape and its derivatives, as shown previously. As themechanical mode shape varies according to change in proofmass properties, these terms change accordingly and thus, it is

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

Figure 3. Connection between the electrostatic shaker (as the part that moves the system), the clamping device, and the energy harvesterdevice with proof mass I. Energy harvester device is electrically wired through electrical leads in order to pull the voltage/current off as shownin the left.

also necessary to evaluate the mechanical damping ratio thatis appropriate to each proof mass (i.e., device). As dampingis typically measured at the device natural frequency, which isfixed, the damping dependence on frequency is not consideredhere. However, if the damping ratios are not the same atthe resonance and anti-resonance frequencies, the dependenceof the mechanical damping ratios on frequencies should beexamined.

3. Experimental procedures

3.1. Experimental setup and device performancemeasurements

A macroscale, cantilevered plate is used as a test device.The schematic of a bimorph device with addition of a proofmass at the end of the device along with electrical connectionis shown in figure 2. A brass reinforced bending actuator(Piezo Systems Inc., T226-A4-503X) is composed of sevenlayers: two piezoelectric (active) layers, each of which issandwiched by two electrode layers (four layers of electrodesin total) and one structural (inactive) layer made of brass. Twopiezoelectric layers (PZT-5A) are poled in opposite directionsso that opposite strains in each layer generate electric fieldsin the same directions. For bimorph configuration, there aretwo ways to interconnect active layers: series and parallelconnections. Series connection is chosen to match the polingdirection in use. Further details on poling direction andrelevant coordinate systems are described in [22–24]. The{3-1} operation is employed where the strain develops in thedirection perpendicular to the electric field. Proof masses areadded at the free end of the piezoelectric mechanical vibrationenergy harvester using superglue and insulating tape. In orderto mount the energy harvester device on an electrostatic shakerfor input base excitation, a simple aluminum clamp is adopted,which is shown in figure 3.

Absolute tip displacement as well as the base displacementin transverse direction is measured to obtain the relativemechanical displacement using a laser Doppler vibrometer(Polytec PSV 300-H). As illustrated in figure 2, a simple circuit

that consists of purely resistive electrical load in conjunctionwith a bimorph device electrically connected in series isestablished to achieve electrical responses of the device. Apurely resistive electrical load is helpful not only to simplifythe calculation but also to measure the power generated. Themeasured voltage is root-mean-squared voltage, vrms, and isconverted to vmax by multiplying by

√2. Power is evaluated

from the measured voltage and applied electrical resistance,Rl, through the relation, Pmax = v2

max/Rl. To determine theoperating points, it is important to assess both resonant andanti-resonant frequencies, which is accomplished by sweepinga range of frequencies using a laser vibrometer either at short-circuit conditions or at open-circuit conditions. The resonantfrequency is defined as the natural frequency when the device isin the short-circuit condition while the anti-resonant frequencyis the natural frequency in open-circuit condition. Anti-resonant frequency, in particular, is an important measure ofelectromechanical coupling and thus, from the information ofanti-resonance frequency, it is possible to assess the extentof electromechanical coupling in the system. Once resonantand anti-resonant frequencies are found, both mechanical andelectrical tests are performed at resonances, near resonances,and away from resonances with varying electrical loadingconditions. The results are graphically presented in the nextsection in comparison with simulated results from analyticalmodeling.

3.2. Test device dimensions and material properties

Geometric parameters of the devices and material propertiesused in model implementation are listed in table 1. Dependingon availability, values are adopted from publications,measured, or calculated. It is necessary to differentiate energyharvester length, LEH, from actual energy harvester length,L tot. Actual energy harvester length describes the length ofthe energy harvester plate excluding the length occupied bythe clamping device. As the thickness of nickel electrode isabout 1 μm, it is regarded as negligible considering the entiremacroscale device. Regarding material properties, severalvalues such as density, elastic stiffness, piezoelectric constant,

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

Table 1. Geometric and material properties of the tested macroscale bimorph energy harvesters.

Value Comment

Device dimensions

Energy harvester length LEH (mm) 63.7 MeasuredCantilevered energy harvester length L tot (mm) 53.0 MeasuredEnergy harvester width b (mm) 31.7 MeasuredEnergy harvester thickness t (mm) 0.675 MeasuredEnergy harvester mass MEH (g) 10.5 MeasuredPiezoelectric layer thickness tp (mm) 0.275 MeasuredStructural layer thickness ts (mm) 0.126 Measured

Material properties used

Piezoelectric layer density ρp (kg m−3) 7750 Reference [32]Structural layer density ρs (kg m−3) 7630 CalculatedProof mass density ρ0 (kg m−3) 7751 CalculatedPiezoelectric layer stiffness cE

11 (GPa) 61 Reference [32]Structural layer stiffness cs (GPa) 100 CalculatedAbsolute permittivity εT

33 (F m−1) 1800ε0 Reference [33]

Absolute permittivity εS33 (F m−1) 1551ε0 Calculated

Vacuum permittivity ε0 (F m−1) 8.854 × 10−12 Reference [34]Strain constant d31 (10−12 C m−1) −190 Reference [33]Strain constant e31 (C m−1) −14.2 Calculated

Table 2. Geometric dimensions of device proof masses and input base accelerations.

Mass,M0 (g)

Active platelength, L(mm)

Proof masslength, L0

(mm)

LTOT

(=L + L0)(mm)

Proof massthickness, t0

(mm)

Baseacceleration(m s−2)

No proof mass 0 53.0 0 53.0 0 2.5Proof mass I 16.7 42.7 10.4 53.1 6.61 0.5Proof mass II 34.7 31.6 21.6 53.2 6.66 0.2

and the absolute permittivity at constant stress of piezoelectriclayer (PZT-5A) are taken from available literature. Both thedensity of structural layer and proof masses are calculated fromthe known values of mass and volume. The elastic stiffnessof the structural layer is found to be 100 GPa (inferred fromdevice resonant frequency, as it was not directly measured).While piezoelectric strain constant, d31 is available as −190 ×10−12 (C m−1), the stress constant, e31 is computed using itsrelation with coupling coefficient matrix, �. Refer to [23] formore detailed description of measurement methods for eachproperty. For plate structures, the absolute permittivity atconstant strain, εS

33, is related to the absolute permittivity atconstant stress, εT

33, through εS33 = εT

33 − 2d31e31, and thus, theabsolute permittivity at constant strain is evaluated based on theknown information of εT

33, from publication and piezoelectricconstants. Masses, lengths, thicknesses of two different sizesof proof masses are presented in table 2 together with inputbase accelerations at which the energy harvester device with aproof mass is excited.

4. Experiment-model correlation and discussion

4.1. Model implementation

Model simulations are run on the macroscale energy harvesterwithout a proof mass (no PM), with proof mass I (PM I),and with proof mass II (PM II), followed by experimental

performance tests for the purpose of model verification. Itis beneficial to test the device without a proof mass as theexperimental results can be compared with prior work by duToit et al [25], where the same type of device was testedwithout a proof mass. Key effective parameters such as mass,stiffness, capacitance, etc that appear in governing equationsfor model implementation are listed in table 3. Overall, alleffective parameters in the governing equations (38) and (39)vary depending on whether a proof mass is added or not aswell as geometric specifics of the proof mass. Consistent withexpectations, the effective mass, M , increases as the size andthe mass of a proof mass increases and that effective stiffness,K increases as the active beam/plate length gets shortened dueto longer proof mass, when proof mass I (PM I) and proof massII (PM II) are compared. Other parameters such as coupling,θ , capacitance, Cp, and system coupling, κ2, are found notto change significantly with proof masses as they are onlyaffected by coupling constant and electrical mode shapes, notmechanical or geometric factors. Mechanical damping ratiois also obtained following the procedure stated in section 2.5previously, where mechanical responses measured at short-circuit condition are required along with theoretical valuesof effective mass, M , forcing function, Bf, and mechanicalmode shape at the tip of the device, ψr,tip. It should benoted that the mechanical mode shape at the tip, ψr,tip, aswell as effective mass, M , and forcing function, Bf, increasesignificantly as proof mass becomes larger and heavier. It

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

(a) (b)

Figure 4. Variation of mechanical mode shapes and second derivatives (strain is proportional to the second derivative) in the axial directionfrom the clamp (x = 0) to the proof mass junction (x = 1). (a) ψr , (b) second derivative of ψr .

Table 3. Key device parameters for model implementation.

M (kg) K (N m−1) � (N V−1) Cp (×10−8 F) Bf κ2 ψr,tip ζm

No PM 0.008 78 4 150 −0.004 688 4.194 0.006 872 0.1262 2 0.0182PM I 0.114 7 760 −0.006 653 4.202 0.049 58 0.1367 2.767 0.0154PM II 0.414 19 900 −0.009 614 4.210 0.126 4 0.1110 4.301 0.0146

is important to note that the mechanical damping ratios arederived from the equation of mechanical displacement at short-circuit conditions and resonance and thus computed usingthe data measured at short-circuit condition and resonance.As it is also possible to achieve the mechanical dampingratios utilizing the equations and measured data at open-circuit conditions and anti-resonance as well, they are alsocomputed to confirm that damping has little dependence onfrequency in that range. Although detailed damping analysisat open-circuit condition and anti-resonance is not presentedhere, the key result from comparison is that there is littledifference in the values of mechanical damping ratios analyzedeither at short-circuit and resonance or at open-circuit andanti-resonance. Determination of damping is critical and thescheme suggested and utilized in this work is effective, assmall changes in values of mechanical damping ratios causelarge changes in device response, especially at the resonances.This implies the importance of considering the geometric andmaterial properties of a proof mass when a device incorporatesa proof mass in order to get exact treatment of device dynamics(e.g. damping ratio analysis) instead of using a point massassumption. Key performance resulting equations are used togain mechanical tip displacement histories as well as electricalresponses as a function of electrical loading, Rl, whichare plotted together with experimental results in figures 5through 13.

4.2. Mechanical mode shapes for the proof masses

Mechanical mode shape, ψr , plays a major role in describingthe structural dynamics of the beam/plate harvester. In thisinvestigation, the first vibration mode would generally give thegreatest power, and the subsequent experiment targets the firstvibration mode as the frequency of interest for these energyharvesters, i.e., the input energy at higher modes will be lessthan at the first mode as it is a design choice. Importantly,axial strain, (see equation (8)) is written in terms of the second

derivative of mechanical mode shapes. The importance ofmechanical mode shape should be highlighted when a proofmass is attached at the end of an energy harvester device. Infigure 4, the effect of proof mass on the mechanical modeshape and the second derivatives of mode shapes are notedto vary. The horizontal axes represent x, an axial position ofnormalized length of the device plate, where the 1.0 indicatesthe junction point between the device plate and a proof mass.Figure 4(a) clearly shows that the mechanical mode shape isnormalized to 2 consistently for all cases at the junction wherethe proof mass begins. The plot in figure 4(a) reveals thatthe mechanical mode shapes change slightly, depending onwhether there is a proof mass or not (no PM versus PM I or II),as well as on the size of the proof masses. When the secondderivatives of mechanical mode shapes are compared with oneanother, the difference along the plate length in the valuesof second derivatives of mechanical mode shapes is moreapparent. Considering figure 4(b), in addition to managingresonant frequency, adding a proof mass has two additionalpositive effects: (i) the maximum strain, and therefore stress,is reduced at the cantilever root, and (ii) average strain isincreased due to non-zero strain at the proof mass (device end).

4.3. Performance of piezoelectric vibration energy harvesters

4.3.1. Resonant and anti-resonance frequencies. Our modelin section 2 allows us to predict resonant and anti-resonantfrequencies, which are compared with experimental results.As shown in table 2, the dimensions of PM I and PM IIare distinct particularly in that PM II is longer in the axialdirection, therefore having more distributed mass over theenergy harvester than PM I. As they have similar thickness,this geometric sequence is quite beneficial to investigate theinfluence of mass distribution as well as the geometric variationon the device of rigid body proof masses. Table 4 summarizesresonant frequencies and anti-resonant frequencies obtainedfrom simulation and measurement for all proof mass cases.

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

(a) (b)

(c) (d)

(e) (f)

Figure 5. No proof mass (no PM): predicted versus measured response: tip displacement plotted against the electrical load at various inputfrequencies. Base acceleration is held constant at 2.5 m s−2. Empty squares in red color indicate the tip displacements measured atopen-circuit condition (Rl → ∞). (a) 75 Hz, (b) 95 Hz, (c) 109.5 Hz (resonance), (d) 115.25 Hz (anti-resonance), (e) 135 Hz, (f) 160 Hz.

Table 4. Summary of frequencies, dimensionless time constants and electrical load both at short-circuit condition and open-circuit condition.

Calculated Calculated

CaseMeasuredf1,r (Hz)

Calculatedf1,r (Hz)

Measuredf1,ar (Hz)

Calculatedf1,ar (Hz) αr,opt αar,opt Rr,opt (k ) Rar,opt (k )

No PM 109.5 109.45 115.25 116.15 0.278 3.21 9.62 111.5PM I 41.63 41.44 44.5 44.16 0.222 3.98 20.3 364.0PM II 34.75 34.85 37.0 36.73 0.255 3.55 27.6 384.6

Overall, measured values and simulated values of bothresonant frequencies ( f1,r) and anti-resonant frequencies ( f1,ar)are in good agreement for all cases. Considering thefact that an accuracy of down to 0.125 Hz is possible toobtain experimentally, the difference between the measuredfrequencies and calculated ones correspond quite accurately.An important point is that the resonant frequency is greatlyreduced when a proof mass is introduced. For example,the natural frequency at short-circuit condition of the devicewithout a proof mass is measured as 109.5 Hz, which isdecreased by more than half to 41.63 Hz by adding PM I atthe end of the device. Although PM II is roughly twice longerthan PM I, the resonant frequency of PM II (34.75 Hz) is not

lower by half than that of PM I (41.63 Hz). The same trends areseen in not only resonance frequencies but also anti-resonancefrequencies, both empirically and theoretically. The resultsshown in table 4 suggest that the model is capable of predictingboth resonance and anti-resonance frequencies both with andwithout a proof mass.

4.3.2. Model-experiment comparison of overall energyharvester response. Mechanical and electrical deviceresponses are compared here. For each case (No PM,PM I, and PM II) depending on proof masses, mechanicaltip displacement is measured at various discrete electricalresistances ranging from 0 to 100 k with varying operating

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

(a) (b)

(c) (d)

(e) (f)

Figure 6. No proof mass (no PM): predicted versus measured response: voltage developed, plotted against the electrical load at various inputfrequencies. Base acceleration is held constant at 2.5 m s−2. Empty squares in red color indicate the voltages measured at open-circuitcondition (Rl → ∞). (a) 75 Hz, (b) 95 Hz, (c) 109.5 Hz (resonance), (d) 115.25 Hz (anti-resonance), (e) 135 Hz, (f) 160 Hz.

frequencies. Operating frequencies are selected to betwo frequencies below resonances, at resonant and anti-resonant frequency, and two frequencies above anti-resonancesfor comparison, so that both off-resonant operation andresonant operation are systematically analyzed. For PM IIcase, considering the narrow range of frequencies, onebelow resonance and one above anti-resonance frequenciesare chosen as off-resonant operating frequencies instead oftwo points for each. At the same conditions, electricalperformances are obtained by measuring the voltage generatedacross the electrical resistive loads and calculating theextracted power using measured voltage. All experimentalresults are graphically demonstrated in figures 5 through 13using dots while predicted results from modeling arerepresented by lines on the same plots. In addition to the datapoints from 0 to 100 k , the measured points at open-circuitcondition, where electrical loading, Rl, goes to infinity, areincluded at each input frequency in the plots of mechanical tipdisplacements (figures 5, 8, and 11) and voltages (figures 6, 9,and 12) versus electrical resistance, Rl. Empty squares in redcolor are used to distinguish these points from other measured

points. As power goes to zero at open-circuit, these points arenot considered in power–resistance plots.

Tip displacements are shown in figures 5, 8, and 11, andcompare the experimentally obtained actual tip displacements,wtip, with predicted tip displacement from modeling for noPM, PM I and PM II, respectively. On the whole, thesimulation is in good agreement with experimental results atvarious electrical loadings and multiple operating frequencies.Particularly, at off-resonant operations, for example, at 75 Hzor 160 Hz in figures 5(a) and (f), respectively, the modelaccurately predicts the magnitudes and trends of mechanicaltip displacement histories. It is also shown that as furtheraway from resonance or anti-resonances, the closer the modelis to the experimental results. Regardless of whether thereis an attachment of a proof mass or not, similar trends in tipdisplacement histories result. At frequencies below resonance,tip displacements decrease slightly as the electrical loadingincreases (e.g. figures 5(a), (b), 8(a), (b) and 11(a)). Asthe operating frequency gets closer to resonance, the extentof decrease becomes larger although it does not decreaseproportionally to resistance. For frequencies at the point

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

(a) (b)

(c) (d)

(e) (f)

Figure 7. No proof mass (no PM): predicted versus measured response: extracted power plotted against the electrical load at various inputfrequencies. Base acceleration is held constant at 2.5 m s−2. (a) 75 Hz, (b) 95 Hz, (c) 109.5 Hz (resonance), (d) 115.25 Hz (anti-resonance),(e) 135 Hz, (f) 160 Hz.

of anti-resonance and above anti-resonance, the trends oftip displacements are different in that they increase withincreasing resistance. It should be noted that the maximumtip displacements are obtained at resonance for the short-circuit condition (Rl = 0), and at anti-resonance for theopen-circuit condition (Rl → ∞). These displacements areapproximately equal, and represent the two points where noenergy is harvested.

Voltages generated are plotted against resistance from0 to 100 k at various operating frequencies. Overall,experimental results of electrical responses of the devicesrepresented by voltages agree well with the simulations forall no PM, PM I, and PM II, as shown in figures 6, 9,and 12, respectively. In a similar fashion to the mechanicalrelative tip displacements, when further away from resonancesor anti-resonances, the more precise the predicted valuesare when compared with experimental results. In general,the voltage increases along with the increase of resistancesboth empirically and theoretically at not only off-resonantoperations but also resonant operations. The tendency isthat the voltage is on the sharp increase at relatively low

electrical loadings, which is followed by voltages approachinga saturated point. Although it is not presented here, whenvoltage is plotted in a log–log scale, this trend becomes moreprominent. For voltages, the maximum values are obtainedat anti-resonance, not at resonances. For example, withinthe resistance range of experimental measurements used here(0–100 k ), the highest voltage of 6.9 V appears at anti-resonance among all other operating frequencies in case ofa device without a proof mass (figure 6(d)). However, thedata point of 11.1 V measured at the open-circuit condition(Rl → ∞) suggests that the voltage would keep increasing asthe electrical resistance increases until it reaches this value, ifthe measurements were taken at an extended range of electricalload beyond 100 k . This trend appears more prominentparticularly at anti-resonances for all cases (figures 6(d), 9(d),and 12(c)). Unfortunately, the range of electrical resistance forexperimental measurements was limited here to 0–100 k , butthe experimental trends are indicated by the open-circuit datapoints. On the other hand, at resonances (figures 6(c), 9(c),and 12(b)), it is observed that the voltages measured at100 k are already close to the values measured at open-

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

(a) (b)

(c) (d)

(e) (f)

Figure 8. Proof mass I (PM I): predicted versus measured response: tip displacement plotted against the electrical load at various inputfrequencies. Base acceleration is held constant at 0.5 m s−2. Empty squares in red color indicate the tip displacements measured atopen-circuit condition (Rl → ∞). (a) 20 Hz, (b) 30 Hz, (c) 41.625 Hz (resonance), (d) 44.5 Hz (anti-resonance), (e) 55 Hz, (f) 80 Hz.

circuit condition (Rl → ∞), which is quite in contrast to thetrends at anti-resonances where a large numerical differenceis present between resistance data at 100 k and infinity. Insection 2.3, optimal electrical loading conditions for maximumpower at both resonance and anti-resonance are derived andgiven in equations (49) and (50). Optimal electrical resistancesas well as corresponding dimensionless time constant, αopt atresonance and anti-resonance for all three proof mass cases arecomputed and listed in table 4. As shown in table 4, optimalelectrical loading conditions at resonances are within the rangeof 0–100 k , while optimal electrical resistances appear wellabove 100 k at anti-resonances. For example, 384.6 k and364.0 k are calculated for PM I and PM II, respectively,as optimal electrical resistances where maximum power canbe obtained. This is related to the different behavior atopen-circuit condition between resonance and anti-resonance.If the voltage were measured near the optimal points, like400 k , for instance, at anti-resonance, the measured pointsat these points would approach the values measured at Rl →∞. The empirical results of device responses at open-circuitconditions (Rl → ∞) help predict the tendency of voltage

behavior against electrical loading that are beyond the rangeof measurement undertaken in this work. Also, this tendencycorresponds well with the trends of simulated results. Not onlyvoltages but also tip displacements measured at open-circuitconditions (Rl → ∞) do exhibit the similar behavior in thatthe values at open-circuit conditions (Rl → ∞) represent thedevice responses at extended electrical loading conditions asin figures 5, 6, 8, 9, 11, and 12. One other point to noticein these voltage–resistance figures is the large difference involtage between resonance and anti-resonance. This changeis reflected in the voltage equation (46) yielding:

vAR

vR= κ2

2ζm

√1 + κ2 (64)

where vAR and vR represent the voltages at anti-resonanceand resonance, respectively. In fact, from equation (46), thelargest voltage in the system for a constant base acceleration,wB, is obtained at open-circuit conditions (Rl → ∞). Thiscorresponds with the maximum tip displacement for open-circuit conditions mentioned earlier, and together they serveto identify the anti-resonance frequency, ωAR. Also from

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

(a) (b)

(c) (d)

(e) (f)

Figure 9. Proof mass I (PM I): predicted versus measured response: voltage developed, plotted against the electrical load at various inputfrequencies. Base acceleration is held constant at 0.5 m s−2. Empty squares in red color indicate the voltages measured at open-circuitcondition (Rl → ∞). (a) 20 Hz, (b) 30 Hz, (c) 41.625 Hz (resonance), (d) 44.5 Hz (anti-resonance), (e) 55 Hz, (f) 80 Hz.

equations (45)–(47), a check can be made on the electricalcoupling parameter, κ2, by comparing the resonance and anti-resonance frequencies, ωR and ωAR, using the relation,

κ2 =(ωAR

ωR

)2

− 1. (65)

Power extracted from a test piezoelectric vibration energyharvester is acquired using the relation, Pmax = v2

max/Rl

and are presented in figures 7, 10, and 13 together with thesimulated lines obtained from modeling. The unit of powerhere is micro-watts. Again, the model captures accuratelythe trend of power across varying electrical loading conditionsat various input frequencies. In a prior work by du Toitet al [22–25], it was theoretically shown that there are twooptimum operating points in terms of power maximizationand that those two points correspond to resonance andanti-resonance, giving equal values of maximum powerwhen 2ζm/κ

2 � 1. The optimal points of electricalloadings for maximum power at resonance (Rr,opt) and anti-resonance (Rar,opt) are calculated using equations (49) and (50),

respectively, with the results summarized in table 4. As theoptimal electrical resistances particularly at anti-resonancesexceed the experimental performance test range (100 k ),simulated powers are re-plotted against a more extended rangeof electrical resistances from 0 to 400 k at resonancesand at anti-resonances for all cases of no PM, PM I andPM II, and presented in figure 14. For example, forthe case of PM I, the maximum power, 166.9 μW, takesplace at around 20.3 k at resonant operation whereas thepower is maximized as 164.5 μW at about 364 k at anti-resonance. For no PM case, 335.2 μW at 9.62 k isobtained at resonance while 327.5 μW is computed at itsoptimal resistance of 111.5 k at anti-resonant operation.Power in PM II case is maximized as 60.5 μW at resonanceand 59.3 μW is obtained at anti-resonance at their optimalelectrical resistance 27.6 k and 384.6 k , respectively. Itis quite significant that almost equal maximum power can beachieved at distinct electrical loading conditions, depending onwhether a device is operated at resonance or anti-resonance.This also implies the frequency shift according to electricalloading, which is consistent with the discussion by du Toit

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

(a) (b)

(c) (d)

(e) (f)

Figure 10. Proof mass I (PM I): predicted versus measured response: extracted power plotted against the electrical load at various inputfrequencies. Base acceleration is held constant at 0.5 m s−2. (a) 20 Hz, (b) 30 Hz, (c) 41.625 Hz (resonance), (d) 44.5 Hz (anti-resonance),(e) 55 Hz, (f) 80 Hz.

et al in [25], using power–frequency ratio plots at variouselectrical resistances. The slight numerical difference betweenthe maximum power at resonance and anti-resonance suggeststhat system coupling, κ2, is not large enough compared to themechanical damping ratio, ζm, to satisfy the approximation that2ζm/κ

2 � 1. As already shown in voltage–resistance plots,the maximum voltage appears at anti-resonance rather thanresonance. Therefore, anti-resonant operation is advantageouswhen high voltage is required with high electrical resistancewhile resonant operation is beneficial for the applicationsthat require high current. This explains why not onlyresonant frequency but also anti-resonant frequency should beconsidered for optimal design.

There are two aspects that should be mentioned regardingthe operation at resonances. First of all, the advantages ofresonant (or anti-resonant) operation for energy harvesting areclearly observable when the resonant operation is comparedwith the off-resonant operation. Both mechanical and electricalperformance of the system is greatly amplified at resonancecompared to the off-resonant operation. Small deviations fromresonance or anti-resonance condition will cause the device

to perform with much diminished effectiveness. In terms ofvoltage, the maximum value is seen to be achievable at anti-resonance conditions. Thus, depending on the application,it should be determined whether the systems are operated atresonance or anti-resonance (i.e., if high current or voltageis desired, respectively). Also, adjusting the input conditionssuch that they are as close to the resonance or anti-resonanceis another substantial point when considering the deviceoperations. Secondly, although predicted and measuredperformance results are well correlated both mechanicallyand electrically at off-resonant operations, it is observed thata consistent discrepancy exists at the resonances and anti-resonances between the simulation and the measurement.When prediction deviates from experimental measurementfor voltage, the extent of discrepancy for power is morepronounced because the power is obtained through usingthe value of voltage squared, as shown in power plotsof figures 7, 10, and 13. The deviation that occurs atresonance and anti-resonance is attributed to the nonlinearity inpiezoelectric coupling that is not incorporated in the modeling.The model here is based on a linear constitutive relation

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

(a) (b)

(c) (d)

Figure 11. Proof mass II (PM II): predicted versus measured response: tip displacement plotted against the electrical load at various inputfrequencies. Base acceleration is held constant at 0.2 m s−2. Empty squares in red color indicate the tip displacements measured atopen-circuit condition (Rl → ∞). (a) 25 Hz, (b) 34.75 Hz (resonance), (c) 37 Hz (anti-resonance), (d) 50 Hz.

(a) (b)

(c) (d)

Figure 12. Proof mass II (PM II): predicted versus measured response: voltage developed, plotted against the electrical load at various inputfrequencies. Base acceleration is held constant at 0.2 m s−2. Empty squares in red color indicate the voltages measured at open-circuitcondition (Rl → ∞). (a) 25 Hz, (b) 34.75 Hz (resonance), (c) 37 Hz (anti-resonance), (d) 50 Hz.

of piezoelectricity. However, the piezoelectric constants areknown to be nonlinear even at moderate levels of strain(e.g. 100 μ-strain). Accordingly, a moderate discrepancytakes place at resonant and anti-resonant conditions due tothe nonlinearity and this is consistent with modeling of thenonlinearity by others [35–38]. Although there are smallquantitative differences between measurement and simulation

at the resonances, the general trends of the tip displacement,voltage, and power at resonance as well as at away fromresonance are predicted accurately.

Lastly, the relation between the electrical damping ratioand mechanical damping ratio can be analyzed using tipdisplacement versus electrical loading result. As statedearlier, the electrical damping ratio is known to be equal to

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

(a) (b)

(c) (d)

Figure 13. Proof mass II (PM II): predicted versus measured response: extracted power plotted against the electrical load at various inputfrequencies. Base acceleration is held constant at 0.2 m s−2. (a) 25 Hz, (b) 34.75 Hz (resonance), (c) 37 Hz (anti-resonance), (d) 50 Hz.

mechanical damping ratio when the power is maximized atoptimal electrical loadings. As electrical resistance is zeroat short-circuit condition, no electrical damping does exist atthis point. However, when electrical resistance is introduced,more and more electrical damping affects the device responses.This is apparently shown in the measurement results of tipdisplacement against electrical loading especially at resonance.For PM I case, the maximum power in measurement occurs at20 k at resonant operation (41.625 Hz) as in figure 10(c).If we compare the tip displacement at 20 k , where themaximum power takes place, with the tip displacement atshort-circuit conditions (Rl = 0 k ), the former comes as138.3 μm, which is about a half of the latter, 280.3 μm. It canbe easily derived from equation (45) that the tip displacementat resonance is inversely proportional to mechanical dampingratio, ζm at short-circuit conditions (Rl = 0 k ). Sincethe electrical damping ratio becomes equal to the mechanicaldamping ratio at the point where the power reaches maximum,the total damping at the point of maximum power will be twotimes the total damping at short-circuit condition where onlya mechanical damping is involved. This coincides with theobservation on the decrease in experimentally measured tipdisplacements by about a half when the value at short-circuitcondition is compared with the data measured at the pointwhere power is maximum.

The electromechanically coupled energy harvester modelpredicts quite accurately both mechanical and electricaldevice responses overall, especially at off-resonant operationsboth below and above resonance or anti-resonance. Themodel has a capability of not only capturing the generaltrend of system response but also estimating quite preciselyvalues of magnitude for resonance frequencies, mechanicalperformances, and electrical device responses. The governingequations of the system (equations (56) and (57)) are almost

identical to those used by Erturk et al [28], but they used mass-normalized modes and complex quantities, which obscure themeaning of the resulting calculations; although the proof masswas not rigorously treated, the behavior in that work doesindicate (undiscussed) resonant and anti-resonant behavior.

5. Conclusions and recommendations

An analytical model for a cantilevered piezoelectric vibrationenergy harvester focusing on the addition of a proof massis presented. Material and geometric properties of a proofmass as well as the cantilevered beam/plate structures arethoroughly considered in the coupled electromechanical modelby analyzing the exact vibration and mode shapes of thesystem. The system has been nondimensionalized to show keyparameters such as resonant frequency, mechanical dampingfactor, system piezoelectric coupling, and nondimensionalelectrical loading (ω1, ζm, κ2, and α, respectively). Thisalso helps discern the behavior of the system easily throughthe resulting equations (equations (45)–(47)). The modelis experimentally verified using a macroscale, symmetric,bimorph, cantilevered PZT bimorph device operating in{3-1} mode with and without a proof mass. Experimentaltests undertaken on the energy harvesting device with differentgeometry proof masses demonstrate that the properties of theproof mass affect the resulting performance of the energyharvester device beyond simply reducing the natural frequency.The predicted results from the simulation are in goodagreement with the experimental results of device responsesboth mechanically and electrically. Especially, a schemeto obtain the mechanical damping ratio from modeling andexperiments is investigated and found to contribute to accurateprediction of device performances. In this work, a distributed

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Smart Mater. Struct. 19 (2010) 045023 M Kim et al

(a) (b)

(c) (d)

(e) (f)

Figure 14. Dual-power optimization at resonance and anti-resonance for each proof mass case. (a) No PM: 109.5 Hz (resonance), (b) No PM:115.25 Hz (anti-resonance), (c) PM I: 41.625 Hz (resonance), (d) PM I: 44.5 Hz (anti-resonance), (e) PM II: 34.75 Hz (resonance), (f) PM II:37 Hz (anti-resonance).

rigid proof mass is assumed and thus this will be followedby an analysis on the flexible proof mass as a future work.Also, the importance of anti-resonant operation as well as theresonant operation is emphasized. Consistent with prior workfor cantilevered harvesters without a proof mass, the workhere shows again that the two frequency ratios at resonanceand anti-resonance are the best candidates for optimal deviceoperating points in terms of power maximization. The exactanalysis of a proof mass in an energy harvesting device helpsavoid the necessity of choosing an arbitrary point for a pointmass position and thus provides a more accurate predictionboth on frequencies and device performances, depending onthe properties of proof masses. Future work will incorporatestructural and piezoelectric coupling nonlinearities into themodeling [38] so that the moderate discrepancy between thesimulation and the experimental measurement at resonance andanti-resonance can be deduced. The effects of higher orderstructural vibration modes on the overall system response alsoneed further study. The developed analysis can be used as anoptimization tool to both predict device performance of MEMSand other cantilevered harvesters containing a proof mass, aswell as in general design for different applications.

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