Mechanically and electrically nonlinear non-ideal ...

Nonlinear Dyn (2020) 99:625–641https://doi.org/10.1007/s11071-019-05091-6

ORIGINAL PAPER

Mechanically and electrically nonlinear non-idealpiezoelectric energy harvesting framework withexperimental validations

Stephen Leadenham · Alper Erturk

Received: 20 November 2018 / Accepted: 22 June 2019 / Published online: 5 July 2019© Springer Nature B.V. 2019

Abstract In the literature of vibration energy harvest-ing, mechanically nonlinear frameworks have mostlyemployed linear electrical circuitry to formulate ACinput–AC output problems, while the existing effortson nonlinear power conditioning circuits have assumedlinear mechanical behavior. However, even for the sim-plest case of a stiff (geometrically linear) piezoelectriccantilever,material softening anddissipative nonlinear-ities in the mechanical domain have to be accomodatedto accurately predict the response for moderate to highexcitation levels, and likewise a stable DC signal mustbe obtained to charge a storage component in realis-tic energy harvesting applications. Furthermore, oftentimes the voltage output is not large enough to assumeideal diode behavior to reduce diodes to switches inAC–DC conversion modeling. Therefore, a relativelycomplete representation of piezoelectric energy har-vesting requires accounting for the mechanical (e.g.,material and dissipative) nonlinearities as well as thenonlinear process of AC–DC conversion with non-ideal circuit elements, such as real diodes. To this end,we present and experimentally validate a multiphysics

S. LeadenhamLawrence Livermore National Laboratory,7000 East Avenue, Livermore, California 94550, USAe-mail: [email protected]

A. Erturk (B)The George W. Woodruff School of MechanicalEngineering Georgia Institute of Technology,Atlanta, Georgia 30332, USAe-mail: [email protected]

framework and harmonic balance analysis that com-bines these mechanical and electrical nonlinear non-ideal effects to predict the DC electrical output (DCvoltage across the load) in terms of the AC mechanicalinput (base vibration) for arbitrary vibration and volt-age levels. The focus is placed on a bimorph cantileverwith piezoelectric laminates under base excitation. Theterminals of the piezoelectric layers are combined inseries and connected to a bridge rectifier with non-idealdiodes and a filter capacitor. The multi-term harmonicbalance framework can capture the ripple in theDC sig-nal as well as amplitude-dependent nonlinear dynam-ics accounting for realistic diode behavior. In additionto quantitative comparisons and validations by com-paring the experimental data and model simulations,important qualitative trends are unveiled for mechani-cally and electrically nonlinear non-ideal dynamics ofpiezoelectric energy harvesting.

Keywords Nonlinear ·Vibration · Energy harvesting ·Piezoelectricity · Electromechanical systems

1 Introduction

Research on vibration energy harvesting methods andtechnologies has received growing attention over thelast two decades to enable energy-autonomous low-power systems, such as wireless sensor nodes [1–3].Among the various approaches for converting ambi-ent vibrations into electricity [4–16], piezoelectric

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626 S. Leadenham, A. Erturk

transduction has been most heavily researched espe-cially due to the high power density and ease of appli-cation of piezoelectric materials [17–19].

The most commonly used piezoelectric energy har-vester configuration is a cantilever with piezoceramiclayers located on a vibrating host structure for elec-trical power generation from resonant bending vibra-tions under base excitation. Theoretical and experi-mental aspects of the linear-resonant cantilever designhave been investigated extensively in the existing lit-erature [8–10,20]. Beyond the basic linear dynam-ics, nonlinear effects in piezoelectric energy harvest-ing have received dramatically increased attention inthe last decade. Three types of basic nonlinearitiescan be mentioned: (1) designed mechanical nonlin-earities (e.g., geometric, magnetoelastic, etc.) for sub-stantial modification (or distortion) of the linear reso-nance via nonlinear effects and bifurcations [21,22]to enable broadband energy harvesters, leading toa number of efforts to create nonlinear energy har-vester configurations (e.g., monostable and bistableDuffing oscillators) via intentionally introduced non-linearities [23–31]. (2) Additionally, the piezoelec-tric materials themselves display appreciable inher-ent nonlinear constitutive and dissipative behavior formoderate to large excitation intensities [32–38]. (3)Thirdly, in order to convert the alternating current pro-duced by a vibrating piezoelectric energy harvesterto a form useful for powering electronic devices orcharging electrical storage units, rectifiers and reg-ulators must be used to produce a stable voltageand direct current output [39], and other nonlinearcircuit opportunities involve nonlinear processing ofthe piezoelectric output, e.g., via voltage inversion[40].

In the literature of piezoelectric energy harvesting,mechanical and electrical nonlinear effects have sofar predominantly been studied separately. Mechani-cally and electromechanically nonlinear modeling andanalysis frameworks [32–38] have employed linearelectrical circuitry to formulate AC input–AC out-put problems, while the existing efforts on modelingand analysis of nonlinear circuits have assumed linearmechanical behavior [41] or idealized rectifier mod-els [40,42,43] that simplify diodes to switches. It isthe main goal of this work to explore the combinedpresence and interactions of inherent mechanical andbasic non-ideal circuit nonlinearities through rigorousmodeling and experiments within a complete, unified

harmonic balance analysis framework. In the follow-ing, a lumped-parameter model of the nonlinear reso-nant behavior of a vibration energy harvester connectedto a bridge rectifier (with non-ideal diodes), filter capac-itor, and load resistor is derived. The energy harvesterdesign of interest is a stiff (hence geometrically linearbut materially nonlinear) piezoelectric bimorph can-tilever with piezoelectric layers connected in series.The problem is explored first to characterize mechani-cal nonlinearities by exploring the AC input (base exci-tation) and AC output (voltage across the load) prob-lem in the presence of a resistive load. Linear modelparameters are validated for experiments in the linearregime. Material softening and dissipative nonlineari-ties are identified from nonlinear tests at different exci-tation levels. Having characterized themechanical non-linearities, the AC input–DC output case is studied. Forthe multiphysics equations of the fully coupled non-linear system, a harmonic balance solution is appliedto analyze the mechanically and electrically nonlinearsystem dynamics at different excitation levels. Experi-mental validations are presented, and quantitative andqualitative trends are discussed.

2 Governing nonlinear electromechanicalequations

Detailed treatment of the derivation for an electrome-chanical model of a geometrically linear (stiff) andmaterially nonlinear bimorph piezoelectric energy har-vester and its harmonic balance analysis can be foundelsewhere [37] (that work provides a detailed accountand justification of the fact that piezoelectric soft-ening is dominantly quadratic, rather than cubic).The piezoceramic cantilever in the experiments ofthis work is very stiff; hence, the geometric non-linearities (stiffness hardening and inertial soften-ing) are negligible, whereas material nonlinearity issignificant; therefore, the aforementioned modelingapproach [37] is suitable (note that geometric nonlin-earity can easily be accomodated as needed [38]). Afterreducing the governing partial differential equationsinto a lumped-parameter form, the differential equa-tions governing the mechanical and electrical behav-ior of the energy harvester forced by base excita-tion (Fig. 1) near a resonance frequency are givenby:

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Mechanically and electrically nonlinear non-ideal piezoelectric energy 627

Fig. 1 Schematic of a cantilevered bimorph piezoelectric energyharvester under base excitation (resultant terminals of the elec-trodes can be connected to a linear/nonlinear circuit)

mx + b1 x + b2 x2 sgn(x) + k1x

+ k2x2 sgn(x) − θvp = − ma(t), (1)

C p vp + i p + θ x = 0, (2)

where x is the relative transverse displacement of thebeam tip with respect to the base, vp is the elec-trode voltage, i p is the electrical current flowing outof the energy harvester, and a(t) is the prescribedbase acceleration. The effective beam mass, m, first-and second-order dissipation coefficients, b1 and b2,first- and second-order stiffness coefficients, k1 and k2,electromechanical coupling parameter, θ , forcing mass(due to base excitation), m, and capacitance, C p, aredefined in the same way as in [37].

This model includes nonlinearities in the mechani-cal dissipative and restoring forces (due to b2 and k2),but assumes linear electromechanical coupling behav-ior. As was shown in [37], electromechanical couplingnonlinearities are less important than the dominantmechanical nonlinearities for the electric field levelsin energy harvesting. To complete the model, a consti-tutive relationship between the electrode voltage, vp,and the harvester output current, i p, must be defined.The harvester electrodes may be shunted across a loadimpedance, yielding a linear dynamic constitutive rela-tionship. This allows for the mechanical nonlinearitiesto be identified and quantified separately from any cir-cuit nonlinearities. For practical use, however, the alter-nating current produced by the piezoelectric cantilevermust be converted to direct current. This process cannotbe implementedwith linear circuit elements and sowillyield a nonlinear constitutive relationship. The follow-ing sections deal with modeling the energy harvesterconnected first to a simple load resistance to identifythe mechanical dissipative and elastic nonlinearities,and secondly connected to a practical harvesting cir-cuit using a full-wave diode bridge for rectification.

3 Harvester connected to a load resistance

To study and quantify the mechanical dissipativeand elastic nonlinearities contributing to the vibratorybehavior of the energy harvester, the electrodes areshunted across a load resistance. This results in the sim-plest possible constitutive relationship between elec-trode voltage and current, namely the algebraic equa-tion given by Ohm’s law:

vp = Ri p. (3)

Choosing the electrode voltage to be the electrical statevariable and substituting in the above current balanceequation yields:

C p vp + 1

Rvp + θ x = 0. (4)

At low response amplitudes, the dissipative and elas-tic nonlinearities of this model will disappear, and thesteady-state behavior of the system can be found usinglinear systems theory. Derivations of steady-state solu-tions in the linear regimewill be given first, followed bypreparation of the equations for solution by the methodof harmonic balance.

3.1 Linear problem

If response amplitudes are restricted to be small,second-order terms in the governing equations can beneglected, yielding the following linear model for apiezoelectric bimorph cantilever excited by transversebase acceleration with electrodes shunted across a loadresistance:

mx + b1 x + k1x − θvp = − ma(t) (5)

C p vp + 1

Rvp + θ x = 0. (6)

To find the complex frequency response functions,which describe the amplitude gain and phase shift of theresponse signals to harmonic excitation, the followingis assumed,

a(t) = Ae j�t (7)

x(t) = Xe j�t (8)

vp(t) = Vpej�t . (9)

where j is the unit imaginary number. Substituting theassumed solutions in the linear system and solving for

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the complex amplitude ratios yields the solutions forthe relative displacement and voltage responses for thevibration harvester connected to a load resistance.

X

A= −m( j�RC p + 1)

(−�2m + j�b1 + k1)( j�RC p + 1) + j�Rθ2

(10)

Vp

A= j�m Rθ

(−�2m + j�b1 + k1)( j�RC p + 1)+ j�Rθ2

(11)

If dielectric losses are non-negligible and the dielec-tric loss tangent, tan δe, of the piezoelectric material isavailable, the capacitance, C p, may be replaced by acomplex capacitance, C p, defined by:

C p = Cp (1 − j tan δe) . (12)

Similarly, if the harvester electrodes are shunted acrossa more complicated linear circuit, the load resis-tance, R, need only be replaced by a general com-plex load impedance, Z(�). Additionally, the kine-matic response signal measured in experiments is thecantilever tip velocity in the inertial frame rather thanrelative tip displacement. Modifying the frequencyresponse functions to include dielectric losses and cor-respond to experimentally measured signals yields thefollowing expressions, with X I representing the com-plex velocity amplitude measured in the inertial frame.

X I

A= − j�m( j�RC p + 1)

(−�2m + j�b1 + k1)( j�RC p + 1) + j�Rθ2

+ 1

j�(13)

Vp

A= j�m Rθ

(−�2m + j�b1 + k1)( j�RC p + 1)+ j�Rθ2

(14)

By comparing these to experimentally generated fre-quency response functions, the quality of first princi-ples predictions (for low amplitudes) can be evaluated,and model parameters can be updated.

3.2 Mechanically nonlinear, electrically linearproblem

To study the steady-state behavior of the governingequations for moderate to large response amplitudes

in a stiff cantilever (for which geometric nonlineari-ties are negligible), material nonlinearities in the formof the second-order dissipation and stiffness terms areretained, and the excitation due to the base accelerationis restricted to be periodic. As is commonly done, thecase of pure harmonic excitation is examined for sim-plicity. Because the system is nonlinear, superpositionno longer holds, and therefore, the governing equationsmust be real valued.To includedielectric loss, the imag-inary portion of the current flowing through the capac-itor due to the loss tangent is recognized to act like anadditional parasitic resistance in parallel with the loadresistance. The governing equations for the nonlinearbehavior of a piezoelectric bimorph excited by a har-monic base acceleration with electrodes connected toa load resistance can therefore be expressed as:


+ k2x2 sgn(x) − θvp = − m A cos(�t) (15)

C p vp +(

�C p tan δe + 1

R

)vp + θ x = 0 (16)

Nonlinear differential equations can be simulated usingnumerical integrators, but the presence of transientresponses can make numerical integrators slow to findsteady-state behavior, especially when the system islightly damped. Numerical methods that find periodicsteady-state solutions directly, while accounting forhigher harmonics, are therefore preferable (e.g., theharmonic balance method). To prepare the governingequations for numerical simulation of any kind, it isuseful to put the system in state space form and isadvantageous to nondimensionalize it. To this end, thenondimensional state space system is defined as:

u′ = f(τ,u), (17)

where u is the nondimensional state vector and ()′denotes the derivative with respect to nondimensionaltime, τ . The system is nondimensionalized by definingcharacteristic time, length, and voltage scales (Tc, Lc,and Vc) accordingly.

t = Tcτ, vp = Vcu3, vp = Vc

Tcu′3,

x = Lcu1, x = Lc

Tcu2, x = Lc

T 2c

u′2.

(18)

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Substituting these in the governing equations yields:

mLc

T 2c

u′2 + b1

Lc

Tcu2 + b2

(Lc

Tcu2

)2

sgn(u2) + k1Lcu1

+ k2 (Lcu1)2 sgn(u1) − θVcu3 = −m A cos(�Tcτ)

(19)

C pVc

Tcu′3 +

(�C p tan δe + 1

R

)Vcu3 + θ

Lc

Tcu2 = 0,

(20)

which can be rearranged into state space form to give:

f1 = u′1 = u2

f2 = u′2 = −

(k1T 2

c

m

)u1 −

(k2LcT 2

c

m

)u21 sgn(u1)

−(

b1Tcm

)u2 −

(b2Lc

m

)u22 sgn(u2)

+(

θVcT 2c

mLc

)u3 −

(m AT 2

c

mLc

)cos(�Tcτ)

f3 = u′3 = −

(θ Lc

C pVc

)u2

−(

�Tc tan δe + TcRC p

)u3 (21)

For the method of harmonic balance, the Jacobianmatrix of the system is required, whose elements aredefined by the expression:

Ji j = ∂ fi

∂u j. (22)

Details and examples of implementing a multi-termharmonic balance solution (obtaining the nonlinearalgebraic equations by Galerkin’s weighted residualminimization and then applying amultivariateNewton-Raphson scheme) in energy harvesting problems can befound elsewhere [31,37].

4 Harvester connected to a rectification circuitwith non-ideal diodes

For practical energy harvesting purposes, a stable DCoutput is required to charge a storage component or topower an electronic device. Since a vibration energyharvester naturally produces AC, the output currentmust be rectified and conditioned. The simplest pas-sive way of accomplishing this is with a diode bridge

Fig. 2 Circuit diagram of piezoelectric energy harvester con-nected to a bridge rectifier, filter capacitor, and load resistance.Currents andnodevoltages used formodel derivations are labeled

and filter capacitor. A diagram of a piezoelectric energyharvester connected to such a circuit is shown in Fig. 2.

On the left side of the bridge is the piezoelec-tric energy harvester modeled by a dependent currentsource coupled to the structure motion and the capac-itance of the piezoelectric material. On the right-handside of the diodebridge are thefilter capacitor and a loadresistance. Since for practical purposes theDCpower isthe quantity of interest, and the filter capacitor is prop-erly large, any reactance of the load can be neglected.Node voltages and currents needed to derive the gov-erning equations are labeled. By applying Kirchhoff’scurrent law, the following expressions relating currentscan be found.

i1 = − θ x − C p(v1 − v2) = ia − ic = id − ib (23)

i2 = Cf v + 1

Rv = ia + ib = ic + id (24)

In order to find the governing equations in terms of nodevoltages, a relationship between the voltage across adiode and the current flowing through it is required.For its combination of simplicity and smoothness, theShockley diode model is used:

iD = Is

[exp

(vD

nVT

)− 1

](25)

The Shockley diode model relates the voltage acrossthe diode, vD, to the current flow, iD. It is parameter-ized by the saturation current, Is, and the product ofthe ideality factor and the thermal voltage, nVT. Thesaturation current is the current that will flow back-ward through the diode when a large reverse voltage isapplied and is on the order of 10−12 A. A real diode hasa reverse breakdown voltage at which large amounts ofreverse current will flow. By using the Shockleymodel,

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it is assumed that the diode bridge is properly chosento avoid breakdown. The thermal voltage is definedas VT = kBT/q (where kB is Boltzmann’s constant,T is the operating temperature, and q is the electroncharge) and is approximately 26mV at room tempera-ture. Using the Shockleymodel, the four diode currentscan be expressed as:

ia = Is

[exp

(v1 − v

nVT

)− 1

]

ib = Is

[exp

(v2 − v

nVT

)− 1

]

ic = Is

[exp

(−v1

nVT

)− 1

]

id = Is

[exp

(−v2

nVT

)− 1

](26)

Since the Schockley model is an algebraic model forthe current–voltage characteristic of a diode, it can beshown that the mean potential on the input terminals ofthe diode bridge must always equal the mean potentialon the output terminals. This yields the relations:

vp = v1 − v2, v = v1 + v2,

v1 = v + vp

2, v2 = v − vp

2,

(27)

which reduces the number of voltage that define thesystem from three to two, being the voltage across theelectrodes of the piezoelectric cantilever, vp as before,and the voltage across the load resistance, the nega-tive terminal of which is grounded. Substituting theexpressions for the diode currents and node voltages,v1 and v2, allows simplification of the current balanceequations. The current balance equation involving thecurrent flowing into the diode bridge is given by:

C p vp + θ x + 2Is sinh

(vp

2nVT

)exp

( −v

2nVT

)= 0

(28)

Likewise the current balance equation involving cur-rent flowing out of the bridge is given by:

Cf v + 1

R+ 2Is

[1 − cosh

(vp

2nVT

)exp

( −v

2nVT

)]= 0 (29)

Combining these current balance equations with thedifferential equationgoverning themechanical behavior

above yields a model describing the dynamics ofa piezoelectric cantilever undergoing harmonic baseacceleration excitationwith electrodes connected to theinput terminals of a diode bridge, the output terminalsof which are connected to a filter capacitor and loadresistor in parallel. As before, dielectric losses in thepiezoelectric material are included.


+ k2x2 sgn(x) − θvp = − m A cos(�t) (30)

C p vp + �C p tan δevp + θ x

+ 2Is sinh

(vp

2nVT

)exp

( −v

2nVT

)= 0 (31)

Cf v + 1

Rv + 2Is

[1 − cosh

(vp

2nVT

)

× exp

( −v

2nVT

)]= 0 (32)

From these equations, one canmakequalitative descrip-tions of the system behavior. First, the equations areheavily biased in favor of positive values of the out-put voltage, v, due to the presence of the exponentialfunctions. Secondly, the equation governing the evo-lution of the piezoelectric electrode voltage, vp, is anodd function of vp due to the hyperbolic sine function,while the equationgoverning the evolutionof the outputvoltage, v, is an even function of vp due to the hyper-bolic cosine function. Both of these observations areconsistent with an intuitive understanding of currentrectification. As before, it is advantageous to nondi-mensionalize and cast the system in state space form.The same nondimensionalization and state representa-tion as in Sect. 3.2 are used,with the addition of a fourthstate representing the output voltage.

v = Vcu4, v = Vc

Tcu′4 (33)

Substituting the nondimensional state definitions in thesystem yields the same result for the mechanical gov-erning equation, but new expressions for the electricalgoverning equations:

C pVc

Tcu′3 + �C p tan δeVcu3 + θ

Lc

Tcu2

+ 2Is sinh

(Vcu3

2nVT

)exp

(−Vcu4

2nVT

)= 0 (34)

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CfVc

Tcu′4 + 1

RVcu4 + 2Is

[1 − cosh

(Vcu3

2nVT

)

× exp

(−Vcu4

2nVT

)]= 0, (35)

which can be arranged into state space form:

f1 = u′1 = u2

f2 = u′2 =

−(

k1T 2c

m

)u1 −

(k2LcT 2

c

m

)u21 sgn(u1)

−(

b1Tcm

)u2 −

(b2Lc

m

)u22 sgn(u2)

+(

θVcT 2c

mLc

)u3 −

(m AT 2

c

mLc

)cos(�Tcτ)

f3 = u′3 = −

(θ Lc

C pVc

)u2 − (�Tc tan δe) u3

−(2IsTcC pVc

)sinh

[(Vc

2nVT

)u3

]

× exp

[−

(Vc

2nVT

)u4

]

f4 = u′4 = −

(2IsTcCfVc

) {1 − cosh

[(Vc

2nVT

)u3

]

× exp

[−

(Vc

2nVT

)u4

]}−

(Tc

RCf

)u4 (36)

The Jacobian matrix is defined as previously.

5 Experimental investigation

To verify the validity of the proposed model for the fullnonlinear dynamics of a practically realized piezoelec-tric vibration harvester, a set of three types of experi-ments are conducted. First, the energy harvester is con-nected directly to a set of load resistances, ranging fromnear short circuit to near open circuit, and excited at lowenough base acceleration levels to remain in the lin-ear behavior regime (linear experiments). By compar-ing the predictions of the linear model to experimentalfrequency response functions, linear model parameterscan be validated/identified. Secondly, the energy har-vester is excited by frequency sweeps at various con-stant base acceleration amplitudeswhile still connectedto the same set of load resistors (AC–AC experiments).By comparing the predictions of the nonlinear mechan-ical model to experimental frequency response curves,

the parameters governing the nonlinearmechanical dis-sipation and nonlinear stiffness can be identified. Third,the energy harvester is connected to a bridge recti-fier, filter capacitor, and load resistance to approximatepractical energy harvesting operation (AC–DC exper-iments). Excitation for the AC–DC experiments is thesame as for the AC–AC experiments. Load resistancevalues are the same for all three experiment types.

5.1 Experimental setup

The piezoelectric energy harvester tested consists ofa PZT-5A piezoelectric bimorph cantilever manufac-tured Piezo Systems, Inc. mounted in a custom fixturewith conductive jaws electrically insulated from the restof the clamp forming the two electrodes. The twopiezo-electric layers are poled in opposite directions and areelectrically connected in series. A microscope photo-graph displaying the central brass substrate and upperand lower piezoelectric laminates is shown as an insetin Fig. 3.

For base excitation experiments, the fixture isattached to the armature of a Brüel and Kjær Type

Fig. 3 Close-up photograph of the experimental setup alongwith an inset showing amicroscopic photographof the edgeview:The brass substrate is the light colored central layer, sandwichedby two PZT-5A piezoceramic layers. The piezoelectric bimorphcantilever is fixed in a custom clamp mounted to the shaker withan attached accelerometer. The upper and lower clamp jaws con-tact the nickel electrode plating on the top and bottom surfacesof the cantilever forming the electrodes. Retroreflective tape isplaced near the tip of the cantilever to facilitate laser Dopplervibrometer measurements

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Fig. 4 Overview photograph of the experimental setup. Thelaser Doppler vibrometer, vibration controller, power amplifier,accelerometer signal conditioner, resistance and capacitance sub-stitution boxes, and breadboard for the rectification circuit areshown

4810 mini shaker powered by an Hewlett Packard6826A power supply and amplifier. Frequency sweepsat constant base acceleration amplitude are conductedusing a SPEKTRA VCS201 vibration control systemusing acceleration feedback. The base acceleration sig-nal is measured using a Kistler 8636C5 piezoelec-tric accelerometer powered and conditioned by a Type5134 power supply and coupler. Load resistance andfilter capacitor values are varied using IET Labs RS-201W and CS-301L resistance and capacitance substi-tution boxes, respectively. Rectification is done using aDiodes Incorporated model KBP202G bridge rectifier.Tip velocity measurements are taken using a PolytecOFV-505 laserDoppler vibrometer andOFV-5000 con-troller. Data are collected using National Instrumentsmodels 9215 and 9223 data acquisition units and Sig-nalExpress software. For linear regime tests, the exci-tation signal sent to the amplifier is generated by an NI9263 analog output module. Photographs of the exper-imental setup are shown in Figs. 3 and 4 .

5.2 Linear regime experiments with AC–AC circuit

Linear regime experiments generate measured fre-quency response functions relating the input base accel-eration to the outputs of cantilever tip velocity mea-sured in the inertial frame and the voltage across theload resistance. The set of load resistances consistsof 13 resistance values ranging logarithmically from1k� to 1M� (covering a broad range between short-and open-circuit conditions). From the measured volt-age frequency response function, current and powerfrequency response functions can be calculated. Theexcitation for the linear regime experiments consists ofa rectangular white noise signal. Multiple averages aretaken. The base acceleration level is kept low to ensuresystem behavior is linear, with an RMS (root meansquare) value of the base acceleration noise signal ofapproximately 0.01g.

Figure 5 shows a comparison between the mea-sured experimental tip velocity and electrode voltage,calculated experimental current and power frequencyresponse functions, and predictions from the linearmodels given above. Experimental data are shown bymarkers and model predictions by curves (in theseand following figures). Each color corresponds to adifferent resistance value, with black markers in theexperimental velocity frequency response plot indicat-ing true short circuit, with a peak near 115 Hz, and trueopen-circuit conditions, with a peak near 120 Hz. Firstprinciples geometric and model parameters are basedon distributed-parameter modeling or identified fromexperiments as required. Geometric and material prop-erties of the bimorph needed for the linear model canbe found in Table 1.

Model predictions match very well with experi-mental data. It is notable that this energy harvestershows two separate maxima in the power generationfrequency response function: one at a small load resis-tance value (near-short-circuit conditions with a lowerfrequency of the peak) and one at a large load resistancevalue (near open circuit with a higher frequency of thepeak). This is characteristic of electromechanical res-onant systems that have a large relative degree of elec-tromechanical coupling and sufficiently low damping[2,42]. In the linear regime, the two power generationmaxima are distinct, but will tend to coalesce for largerexcitation levels aswill be discussed further inSect. 6.1.For an ideal piezoelectric material with no dielectricloss, the two peaks in power generation would be of

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Fig. 5 AC input–AC outputtest data and model in thelinear response regime withvarious load resistancevalues. Experimental datashown with markers andmodel predictions shownwith curves. Black markersin the vibration frequencyresponse show the trueshort- and open-circuitcases

Table 1 Material and geometric parameters for the brass-reinforced PZT-5A bimorph cantilever (Piezo Systems, Inc.model number T226-A4-503X)

PZT density ρp 7800 kg

PZT elastic modulus c11 66 GPa

PZT loss factor γ1 0.021

Coupling e31 −14.4 C/m2

Permittivity ε33 14.2 nF/m

Brass density ρs 8500 kg

Brass elastic modulus cs 100 GPa

Overhang length l 51.9 mm

Total length le 63.7 mm

Width b 31.8 mm

PZT thickness (each) h p 0.267 mm

Brass thickness hs 0.127 mm

the same magnitude. For this energy harvester, dielec-tric loss is significant, and so the peak power point nearopen circuit conditions, which experiences higher volt-ages, is lower in magnitude than the near-short-circuitpeak power point.

5.3 Nonlinear regime experiments

Nonlinear regime experiments consist of frequencysweep tests at constant acceleration levels of 0.1, 0.2,and 0.3g RMS. Tests at each acceleration level were

repeated for the same 13 load resistance values, rang-ing from 1k� to 1M�, used in the linear regime exper-iments. The maximum base acceleration amplitude of0.3g RMS is chosen to keep the cantilever tip deflec-tionwithin the safe limits providedby themanufacturer.This choice of base acceleration level corresponds toa safety factor of approximately two for the maxi-mum tip displacement. The frequency sweep rate foreach test must be slow enough for the test to occur atquasi-steady state. Each data point in the experimentalfrequency response curves is the result of an averageof approximately 100 cycles. Mechanically nonlinearregime experiments are first conducted for the AC–ACcircuit configuration, which is the same as in the linearregime experiments, and then followed by the AC–DCcircuit configuration with rectification. To allow directcomparisons of behavior with and without the electri-cal nonlinearities caused by rectification, the nonlinearmechanical and fully nonlinear test cases are conductedwith the same set of base acceleration amplitudes.

5.3.1 AC–AC circuit configuration: mechanicallynonlinear and electrically linear problem

Plots of experimental data and model predictions areshown in Figs. 6, 7 and 8. RMS tip velocity, RMS volt-age, RMS current, and average power are shown. Col-ors again correspond to the load resistance value andmatch those of the linear regime tests. Model predic-tions are generated using the model given above solved

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Fig. 6 AC–AC test dataand model at 0.1g RMSwith various load resistancevalues. Experimental datashown with markers andmodel predictions shownwith curves


using a general numerical harmonic balance solver. Bycomparing the predictions of the nonlinear mechani-cal model to experimental frequency response curves,the parameters governing the nonlinearmechanical dis-sipation and nonlinear stiffness can be identified. Theparameters defining the nonlinear mechanical behaviorare shown in Table 2.

Again there is good agreement between model pre-dictions and experimental results. As the model is arelatively simple model including only nonlinearitiesin the mechanical dissipation and stiffness, it is most

accurate for lower excitation amplitudes and loses someaccuracy as the excitation amplitude increases. In theseplots, the limits of the vertical axis scale with the exci-tation amplitude. The effect of the nonlinear mechan-ical dissipation can be seen in the decrease in theresponse amplitude relative to the excitation amplitudewith growing base acceleration level. The effect of thenonlinear stiffness can be seen in the shifts of the short-circuit and open-circuit resonant peaks. Short-circuitand open-circuit resonant peaks shift from approxi-

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Table 2 Identified quadratic elastic modulus and loss factor

Second-order modulus cnl −83 TPa

Second-order loss factor γnl 0.15

mately 115 and 120 Hz in the linear regime to 110and 115 Hz at 0.3g RMS, respectively.

5.3.2 AC–DC circuit configuration: mechanically andelectrically nonlinear non-ideal problem

For the AC–DC experiments, the steady-state outputvoltage depends on both the filter capacitance and theload resistance. As in the linear and AC–AC experi-ments, the mean power generation level and resonantfrequency depend on the load resistance. Additionally,the amount of unwanted ripple in the load voltage andcurrent depends on the time constant of the output,namely the product RCf . The ripple factor is definedas:

RF =√

V 2RMS − V 2

DC

|VDC| .

The larger the value of the time constant, RCf , thesmaller the amount of ripple. This is desirable, and so apractical energy harvester would have a value for RCf

significantly larger than the ripple period. Experimen-tally, frequency sweep tests are conducted at quasi-steady state, and so the frequency sweep rate of the

must change inversely proportional to the output timeconstant. Therefore, to ensure that the single sweeprate also used in the AC–AC experiments will be slowenough for all tests without making them unnecessar-ily lengthy, filter capacitor values are chosen for eachof the load resistance values to make the output timeconstant always 0.01 seconds, or approximately twotimes the period of the ripple current. This means thatfor load resistance values ranging from 1k� to 1M�,filter capacitor values vary from 10µF to 10nF.

Plots of experimental data andmodel predictions areshown in Figs. 9, 10 and 11. RMS tip velocity, DC out-put voltage, DC current, and average power are shown.Colors again correspond to the load resistance valueand match those of the linear regime tests and AC–ACexperiments.Model predictions are generated using thefully nonlinear model given above solved using a gen-eral numerical harmonic balance solver. By comparingthe predictions of the fully nonlinear model to experi-mental frequency response curves, the parameters gov-erning the rectification circuit can be identified. Theparameters defining the rectification nonlinearity areshown in Table 3. The values used are typical valuesfor silicon diodes and did not require any updating.

The fully nonlinear model agrees well with experi-mental results to predict trends and behavior of a practi-cally realized piezoelectric vibration energy harvester.Themost important qualitative differences between theAC–AC and AC–DC behavior of the energy harvestercan be seen in the tip velocity and power generation fre-

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Fig. 9 AC–DC test dataand model at 0.1g RMSwith various load resistancevalues. Experimental datashown with markers andmodel predictions shownwith curves

Fig. 10 AC–DC output testdata and model at 0.2g RMSwith various load resistancevalues. Experimental datashown with markers andmodel predictions shownwith curves

quency response curves. In the velocity response, themagnitude of the near-short-circuit resonance is greatlyreduced compared to the near-open-circuit response. Inthe AC–AC experiments, the amplitudes of the veloc-ity response are very similar between near-short-circuitand near-open-circuit condtions. In the power genera-tion response, the curves on the left side of the plot cor-responding to near-short-circuit conditions are reducedin height compared to the curves on the right side corre-sponding to near-open-circuit conditions. As the baseacceleration level increases, the discrepancy betweenthe short- and open-circuit responses decreases. At

0.1g RMS, the near-short-circuit velocity responseamplitude is approximately 64% that of the near opencircuit response. This grows to 79% and 85% at 0.2 and0.3g RMS, respectively. This effect is due to the non-ideal nature of the diode bridge rectifier. As the mag-nitude of the voltage across the diode bridge rises, thediodes act more like ideal switches. Therefore, the effi-ciency of the rectifier grows with both the base acceler-ation amplitude, which increases all response signals,as well as the load resistance, which increases steady-state voltage levels. Interestingly, while the near-short-circuit peak power generation will be higher than the

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Fig. 11 AC–DC output testdata and model at 0.3g RMSwith various load resistancevalues. Experimental datashown with markers andmodel predictions shownwith curves

Table 3 Identified parameters for diodes in the bridge rectifier

Saturation current Is 1 pA

Ideality factor n 1

Thermal voltage VT 26 mV

near-open-circuit power generation for a linear piezo-electric cantilever energy harvester due to dielectriclosses, the near-open-circuit conditions will producemore power for a practical energy harvester with recti-fication at finite base acceleration levels.

6 Discussion of the results and basic trends

6.1 Qualitative performance comparisons and trends

It is of interest to understand the effects of the mechan-ical (elastic and dissipative) and non-ideal electrical(rectification) nonlinearities and their interaction witheach other. As a baseline for comparison, the linearregime AC–AC power generation performance can beused. Shown in Fig. 12 is heat map plot of the experi-mental linear regime power generation performance ofthe energy harvester normalized by the square of thebase acceleration level.

The horizontal and vertical axes correspond to theload resistance value and frequency of base excita-tion, respectively. Clearly visible are the two maxima

Fig. 12 Heat map of experimental AC–AC energy harvestingperformance in the linear regime. Color corresponds to the time-averaged power output of the harvester normalized with respectto the base acceleration level squared

of power generation near the short-circuit and open-circuit loading conditions. The presence of two well-separated peak power points is characteristic of sys-temswith strong electromechanical coupling. In the lin-ear regime, the energy harvester produces a maximumnormalized time-averaged power of approximately 8.3mW/g2, and the color of the heatmap corresponds to thepower output relative to that level. For fair comparisons,the shading in the heat maps for AC–AC and AC–DCperformance in the mechanically nonlinear regime arealso normalized in this way.

Figure 13 shows an array of six heat maps forthe time-averaged power generation. From top to bot-tom, the rows correspond to the three base accel-

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Fig. 13 Heat maps ofexperimental energyharvesting performance inthe mechanically nonlinearregime. From top to bottom,rows correspond to the threetested excitation levels of0.1, 0.2, and 0.3g RMS,respectively. The leftcolumn corresponds to theAC–AC case, and the rightcolumn corresponds to theAC–DC case. Colorcorresponds to thetime-averaged power outputof the harvester normalizedwith respect to the baseacceleration level squared.Color is scaled relative tothat of the linear regimeperformance shown inFig. 12. (Color figureonline)

eration levels of 0.1, 0.2, and 0.3g RMS, respec-tively. The two columns correspond to the two cir-cuit configurations, with the AC–AC configurationon the left, and the AC–DC configuration on theright. All six heat maps are shaded relative to the8.3 mW/g2 maximum normalized mean power gener-ation level of the linear regime performance shown inFig. 12.

The first apparent trend is that normalized powergeneration performance is lower in all sixmechanicallynonlinear regime tests than that of the linear regime,as shown by the darker shades. As was shown withearlier modeling, the dominant mechanical nonlinear-ities are in the dissipation and stiffness. Power genera-tion performance decreaseswith increasingmechanicaldamping, and so the nonlinear mechanical dissipation

reduces the normalized power generation performanceas response amplitudes increase at higher excitationlevels. The stiffness nonlinearity only affects the mag-nitude of power generation slightly. Its primary effectis to shift the location of peak power operating pointsto lower frequencies and higher load resistance values.

Secondly, the power generation performance of theAC–DCconfiguration is significantly lower than that ofthe AC–AC configuration. The AC–DC configurationretains the power generation losses due to the nonlin-ear mechanical dissipation and adds electrical dissipa-tion in the bridge rectifier. An idealized bridge rectifier[42,43] dissipates no power. Ideal diodes act as a per-fect conductor for positive bias voltages (no voltagedrop), and a perfect insulator for negative bias voltages(no current flow). A real diode bridge does dissipate

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power and so reduces the power supplied to the load.Therefore, the linear regime AC–AC power generationperformance can be seen as a global upper limit and themechanically nonlinear AC–AC performance as exci-tation level specific upper bounds.

The first and second observed trends show thatthe nonlinear mechanical dissipation reduces rela-tive power generation performance as excitation levelincreases, and the AC–DC circuit configuration willalways have lower absolute performance than the AC–AC configuration due to electrical power dissipationin the diode bridge. However, unlike the AC–AC case,as the base excitation level increases, the normalizedpower generation performance of the AC–DC config-uration improves rather than declines. While losses inthe diode bridge are unavoidable, and it never acts likethe idealized one-way perfect conductor, the exponen-tial nature of a diode’s I–V characteristic means thatas the amplitude of the voltage across the piezoelectricelectrodes increases, the diode bridge acts more andmore ideally. The contribution to the total loss of thediode bridge relative to mechanical dissipation there-fore decreases with increasing excitation level. For cer-tain choices of piezoelectric energy harvester and cir-cuit components, the normalized power generation willincreasewith increasing excitation level as rectificationbecomesmore ideal and thenwill decrease as nonlinearmechanical dissipation becomes dominant.

Finally, a fourth trend can be seen by examining thelinear and mechanically nonlinear power generationperformance of the AC–AC circuit configuration. Asmentioned earlier, this energy harvester shows two dis-tinct peak power points in the linear response regime,characteristic of a strongly electromechanically cou-pled system.At the lowest base acceleration level (0.1gRMS) of themechanically nonlinear tests, the two peakpower points are still visible. However, as the baseacceleration level grows to 0.3g RMS, the two peaksappear to merge and become a single global maximum.This effect is again due to the nonlinearmechanical dis-sipation. The two high-amplitude responses near-short-circuit and open-circuit conditions are preferentiallyattenuated compared to the lower-amplitude responseat between them. The dip in power generation perfor-mance between the near-short-circuit and near-open-circuit conditions therefore disappears as the excitationlevel increases due to the nonlinear mechanical dissi-pation.

6.2 Electromechanical response waveforms

Oneof the benefits of using themethodof harmonic bal-ance to solve the nonlinear lumped-parameter modelsfor a practical energy harvester is that it allows the quicksimulation of steady-state behavior. From the perfor-mance curves and heat maps, the optimal power gener-ation performance of the energy harvester in the practi-cal AC–DCcircuit configuration at 0.3gRMSoccurs atapproximately 115 Hz, with an optimal load resistanceof around 56k�. Simulating these conditions with themethod of harmonic balance takes much less compu-tation time than a time domain numerical simulation.Figures 14 and 15 show the model predicted wave-forms of the relative tip velocity, piezoelectric elec-trode voltage, and load voltage. These waveforms areproduced from the harmonic balance solution includingfrequency components up to five times the excitationfrequency (i.e., five harmonics were used in the Fourierseries expansion of periodic response forms). Notably,the velocity response appears quite sinusoidal, whilethe piezoelectric electrode voltage, vp, clearly showshigher harmonic content. The load voltage, v, shows anoticeable amount of ripple. As discussed previously,a filter capacitance somewhat smaller than what wouldbe used in practical energy harvesting circuit was usedin this work for experimental reasons. However thisexample shows highlights the ability of a the methoddescribed here to find accurate solutions without mak-ing assumptions like a constant output voltage.

Fig. 14 Representative simulated tip velocity response wave-form at 0.3g RMS base excitation for near optimal load resis-tance of 56k� and peak power frequency of 115Hz

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Fig. 15 Representative simulated output voltage, v (red), andpiezoelectric electrode voltage, vp (blue), response waveformsat 0.3g RMS base excitation for near optimal load resistance of56k� and peak power frequency of 115Hz. (Color figure online)

7 Conclusions

A complete representation of resonant piezoelectricenergy harvesting requires accounting for mechani-cal and dissipative nonlinearities as well as the non-linear process of AC–DC conversion with non-idealdiodes. In this paper, a multiphysics harmonic balanceframework was presented by combining these mechan-ical and electrical nonlinear non-ideal effects to pre-dict the DC electrical output (DC voltage across theload) in terms of the AC mechanical input (base vibra-tion). Focus was placed on a stiff (geometrically linear)bimorph cantilever with piezoelectric laminates con-nected to a full-wave rectifier and a filter capacitor. Theproblem was explored first to characterize mechanicalnonlinearities by exploring the AC input (base excita-tion) and AC output (voltage across the load) prob-lem in the presence of a resistive load. Mechanicalnonlinearities were identified and validated for differ-ent excitation levels. A rectifier comprising non-idealdiodes was then introduced to the system along witha filter capacitor. For the multiphysics equations ofthe fully coupled nonlinear system, a multi-term har-monic balance solution (with 5 harmonics) was appliedto analyze the mechanically and electrically nonlinearsystem dynamics at different excitation levels. A fullset of experiments was conducted, showing trends andinteractions of the material softening and dissipative

nonlinearities and the rectification nonlinearities man-ifested by non-ideal diodes. The multi-term harmonicbalance framework can capture the ripple in the DCsignal (due to finite filter capacitance) and the over-all amplitude-dependent nonlinear dynamics account-ing for realistic diode behavior. The experiments vali-dated the proposed modeling method, and highlightedthe need for simulating the full nonlinear dynamics ofresonant piezoelectric energy harvesters for real worldapplications with realistic circuit components.

Acknowledgements This work was supported by the NationalScience Foundation under Grant CMMI-1254262.

Compliance with ethical standards

Conflict of interest The authors declare that they have no con-flict of interest.

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