Nonlinear Dyn (2018) 92:1935–1945https://doi.org/10.1007/s11071-018-4172-7

ORIGINAL PAPER

Resonant nonlinearities of piezoelectric macro-fibercomposite cantilevers with interdigitated electrodes inenergy harvesting

D. Tan · P. Yavarow · A. Erturk

Received: 25 July 2017 / Accepted: 23 February 2018 / Published online: 10 March 2018© Springer Science+Business Media B.V., part of Springer Nature 2018

Abstract We explore the modeling and analysis ofnonlinear nonconservative dynamics of macro-fibercomposite (MFC) piezoelectric structures, guided byrigorous experiments, for resonant vibration-basedenergy harvesting, as well as other applications lever-aging the direct piezoelectric effect, such as resonantsensing. TheMFCs employ piezoelectric fibers of rect-angular cross section embedded in Kapton with inter-digitated electrodes to exploit the 33-mode of piezo-electricity. Existing modeling and analysis efforts forresonant nonlinearities have so far considered conven-tional piezoceramics that use the 31-mode of piezoelec-tricity. In the present work, we develop a framework torepresent and predict nonlinear electroelastic dynamicsof MFC bimorph cantilevers under resonant base exci-tation for primary resonance behavior. The interdigi-tated electrodes are shunted to a set of resistive electri-cal loads to quantify the electrical power output. Exper-iments are conducted on a set of MFC bimorphs over abroad range of mechanical excitation levels to identifythe types of nonlinearities present and to compare theharmonic balance model predictions and experiments.The experimentally observed interaction of quadraticpiezoelectric material softening and cubic geometrichardening effects is captured and demonstrated by the

D. Tan · P. Yavarow · A. Erturk (B)G.W. Woodruff School of Mechanical Engineering,Georgia Institute of Technology, Atlanta, GA, USAe-mail: alper.erturk@me.gatech.edu

model. It is shown that the linearized version of themodel yields highly inaccurate results for typical baseacceleration levels and frequencies involved in vibra-tion energy harvesting, while the nonlinear frameworkpresented here can accurately predict the amplitude-dependent resonant frequency response.

Keywords Nonlinear · Vibration · Piezoelectricity ·Energy harvesting · Composites

1 Introduction

Piezoelectricity is a reversible process in the form ofthe direct effect (conversion of mechanical strain toelectric charge) and the converse effect (conversion ofelectric potential to mechanical strain) and has beenused in numerous applications ranging from sensingand actuation to vibration control and energy harvest-ing over the past several decades. The most typical useof piezoelectric materials in bending (flexural) modeis through the utilization of the 31-mode with uniformelectrodes. The use of 31-mode in bending has beenwell studied for sensing, energy harvesting, and staticor dynamic actuation for decades [1–13], while the 33-mode has been conventionally used for longitudinal(axial) deformations through the use of piezoelectricstacks and bars [14–19].

It is well known that the 33-mode piezoelectricstrain constant is around 50–100% larger than thatof the 31-mode. The 33-mode was first utilized in

123

1936 D. Tan et al.

bending using Active-Fiber Composite (AFC) struc-tures via the implementation of interdigitated elec-trodes (IDEs) as explored in the early work of Bentand Hagood [20] and Bent et al. [21,22]. The AFCtechnology used piezoelectric fibers with a circularcross section which limited the contact area betweenthe fibers and electrodes, yielding reduced electrome-chanical coupling and high dielectric loss. The Macro-Fiber Composite (MFC) technology [23,24] developedat the NASA Langley Research Center overcame thisissue by using fibers with rectangular cross section.The MFC piezoelectric materials have been used instructural sensing and vibration control [25,26], bio-inspired locomotion [27,28], acoustic wave devices[29,30], morphing- and flapping-wing structures [31–34], and in-air/underwater dynamic actuation or energyharvesting [27,35–41].

Constitutive modeling and experimental character-ization of piezoelectric MFCs have been researchedby several groups. Williams et al. [42–44] presentedan experimentally validated model for equivalent ther-mal expansion and mechanical properties of MFCsusing modified classical mixing rules. Deraemaekeret al. [45] reported rule of mixtures calculations ofthe equivalent linear parameters and compared theircalculations with manufacturers data and experimentalresults. Shahab and Erturk [46] coupled homogenizedlinear constitutivemodelingwith a geometrically linearEuler–Bernoulli electro-elastodynamics framework forenergy harvesting, sensing, and actuation problems.They implemented this linear model for linear under-water bio-inspired actuation and thrust generation [47].Nonlinear modeling of piezoelectric materials to datehas been mostly focused on monolithic piezoelectricmaterials such as the geometrically linear (for stiff andbrittle piezoceramics) and materially nonlinear frame-work by Leadenham and Erturk [10] (see others in theirreferences).

In the present paper, we aim to develop a geometri-cally and materially nonlinear framework for resonantmechanical excitation of MFC bimorph cantileverswith a focus on energy harvesting from base excita-tion. The linear constitutive equations are modified toaccount for piezoelectric softening while capturing thegeometric hardening nonlinearity as well as dissipativeeffects. In the following, the governing nonlinear elec-troelastic equations are derived and then solved usingthe method of harmonic balance. Experimental resultsare compared with model simulations for a range of

base excitation levels and electrical load resistance val-ues, and conclusions are drawn.

2 Nonlinear nonconservative electroelasticequations for an MFC bimorph cantilever underbase excitation

For a piezoelectric material utilizing the 33-mode, theelectric enthalpy density, H , can be given by

H = 1

2cE33S

23 − 1

3γ |S3| S23 − e33S3E3 − 1

2εS33E

23 (1)

where cE33 is the elastic modulus at constant electricfield, S3 is the strain, E3 is the electric field, e33 is thepiezoelectric stress constant, εS33 is the permittivity atconstant strain, and γ is a nonlinear strain coefficient toaccount for ferroelastic softening in PZT-5A [10,48].(These parameters are defined for the thin structure andtherefore reduced from the 3D constitutive equations.)Note that the electromechanical coupling and electricfield nonlinearities are neglected since the focus of thiswork is placed on energy harvesting from mechanicalbase excitation, yielding relatively low electric fields.

The bending strain at any point in an Euler-Bernoullicantilever is given by S3 = −zθ,s , where θ is the angu-lar displacement, z is the distance from the neutral axis,and the subscript s refers to a spatial derivative withrespect to the arc length s. Although the electric field isnon-uniformwithin the piezoelectricmaterial, the elec-tric field across two IDEs can be approximated by E3 ≈− v

Le= − λ

Le, where v is the voltage across the IDEs,

λ is the time derivative of the flux linkage, and Le isthe distance between two IDEs. Therefore, the potentialenergy corresponding to the piezoelectric material is

Up =∫ L

0

[∫ b

0

∫ h

0

(1

2cE33z

2θ2,s − 1

3γ |z| z2 ∣∣θ,s

∣∣ θ2,s

− e33zθ,sλ

Le− 1

2

εS33

L2eλ2

)dzdy

]dx (2)

However, as described in Deraemaeker et al. [45]and Shahab and Erturk [46] and illustrated in Fig. 1, therectangular piezoelectric fibers within an MFC (polar-ized in the x-direction in Fig. 1) are separated by passivelayers of epoxy and then embedded within Kapton filmcontaining interdigitated copper electrodes perpendic-ular to the piezoelectric fibers, creating an array of

123

Resonant nonlinearities of piezoelectric macro-fiber 1937

Fig. 1 MFC bimorph cantilever and a close-up of one of itslayers showing active piezoelectric fibers separated by epoxyand embedded in Kapton film containing interdigitated copper

electrodes (left) and a representative volume element (M × Nrepresentative volume elements exist in an MFC layer) and itseffective dimensions (right)

repeating representative volume elements (RVEs) [46].Each RVE is of width be, length Le, and thickness h p

(as depicted on the right side of Fig. 1) and has mate-rial properties defined by the mixing rules (i.e., rule ofmixtures) formulation [45,46,49]:

cE33,e = νcE33,p + (1 − ν) cE33,m

d33,e = 1

cE33,eνd33,pc

E33,p

εS33,e =[νεT33,p + (1 − ν) εT33,m

]− d233,ec

E33,e (3)

where ν is the volume fraction of PZT within a RVE,εT33 is the permittivity under constant stress, d33 is thepiezoelectric charge constant, and the subscripts p, m,and e correspond to the piezoelectric fiber properties,the matrix (epoxy) properties, and the equivalent prop-erties of the RVE, respectively. The section of theMFCcovered in piezoelectric fibers contains M RVEs in thewidth direction and Na RVEs in the length direction,making the active width bact = Mbe and the activelength Lact = NaLe. Part of the MFC is clamped ina way that sections of the piezoelectric fibers remainmotionless and thus are not strained, so we define theclamped active length as L = NLe.

The MFC bimorph is constructed by bonding twoMFCs back to back and wiring them in parallel, sothe total potential energy in an MFC bimorph can bewritten as:

Up =∫ L

0

(1

2E Ipθ

2,s − 1

3γ B

∣∣θ,s∣∣ θ2,s

−ϑpλ [H(s) − H(s − L)] θ,s − 1

2

Cp

NLeλ2)ds

(4)

where H(s) is the Heaviside step function, and the dis-tributed material properties of the piezoelectric fiberswithin the MFC bimorph are

E Ip = 2M∑

m=1

bec33,e

[(h p + hk

)3 − (hk)3

3

]

B =M∑

m=1

be

[(h p + hk

)4 − (hk)4

2

]

ϑp = 2M∑

m=1

bee33,eLe

[(h p + hk

)2 − (hk)2

2

]

= 2M∑

m=1

e33,ebeh p

Le

(h p

2+ hk

)

≈ 2M∑

m=1

e33,eAehpcLe

Cp = 2M∑

m=1

Na∑n=1

εS33,ebeh p

Le≈ 2

M∑m=1

Na∑n=1

εS33,eAe

Le(5)

123

1938 D. Tan et al.

where h p and hk correspond to the thickness of thepiezoelectric layer and the Kapton layer, respectively,and the effective area Ae is introduced to account forthe approximation in the electric field (i.e., Ae is anequivalent RVE area that corresponds to uniform elec-tric field, identified from measured capacitance).

Equation (4) accounts for the potential energy asso-ciated with all of the RVEs within the MFC bimorph,but because copper has a higher Young’s Modulus thanthe PZT-5A utilized in the MFC, the strain energy inthe Kapton–copper layers is not negligible:

Ukc =∫ L

0

1

2E Ikcθ

2,sds

E Ikc = 2M∑

m=1

be (Ecνc + Ek(1 − νc))

[h3k3

+(2hk + h p

)3 − (hk + h p

)33

](6)

where νc is the volume fraction of copper within theKapton–copper layer (approximately 24%), and thesubscripts c, k, and kc refer to the copper properties, theKapton properties, and the combined Kapton–copperproperties, respectively.

The total potential energy for the entire MFCbimorph can therefore be written as

U =∫ L

0

(1

2E Ieffθ

2,s − 1

3γ B

∣∣θ,s∣∣ θ2,s

−ϑpλ [H(s) − H(s − L)] θ,s − 1

2

Cp

NLeλ2)ds

(7)

where E Ieff = E Ip + E Ikc, and the kinetic energy canbe written as

T = 1

2ms

∫ L

0

[ux,t

2 + (uz,t + ub,t )2]ds (8)

where ms is the structural mass per length of the MFCbimorph and ub is the base displacement.

For an inextensible cantilever, the Lagrangian L =T − U needs to include a Lagrange multiplier, �, toaccount for the inextensibility condition [50–52]:

L = 1

2ms

∫ L

0

[ux,t

2 + (uz,t + ub,t )2]ds

− 1

2�

∫ L

0

[(1 + ux,s)

2 + (uz,s)2 − 1

]ds +

−∫ L

0

(1

2E Ieffθ

2,s − 1

3γ B

∣∣θ,s∣∣ θ2,s

−ϑpλ [H(s) − H(s − L)] θ,s − 1

2

Cp

NLeλ2)ds

(9)

Applying Hamilton’s principle,∫ t2t1

(δL + δWNC)

dt = 0, where the virtual nonconservative work on thestructure includes linear structural damping and elec-trical dissipation over a resistive load (of resistance R),

δWNC = ∫ L0

(−czuz,tδuz − λ

Rδλ

)ds, we find

∫ t2

t1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∫ L

0

⎛⎜⎜⎜⎝

msux,tδux,t + ms(uz,t + ub,t

)δuz,t − czuz,tδuz

−�(1 + ux,s)δux,s − �uz,sδuz,s

+(−E Ieffθ,s + γ Bθ2,ssgn(θ,s)

+ϑpλ [H(s) − H(s − L)]

)δθ,s

⎞⎟⎟⎟⎠ds

+(∫ L

0 ϑpθ,s [H(s) − H(s − L)] ds + Cpλ)

δλ

− ∫ L0

λ

Rδλds

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

dt = 0 (10)

First converting fromangular displacements into x- andz-displacements, and then integrating by parts, Hamil-ton’s principle yields three equations of motion (non-linearities kept to cubic order):

[�(1 + ux,s) + E Ieff

(uz,suz,sss

)− (ϑpλ [H(s) − H(s − L)]

),suz,s

],s

= msux,t t[

�uz,s−E Ief f(uz,sss+uz,su2z,ss

)+2γ B∣∣uz,ss ∣∣ uz,sss

+(ϑpλ [H(s) − H(s − L)]),s

(1 + ux,s

)]

,s

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Resonant nonlinearities of piezoelectric macro-fiber 1939

czuz,t − msub,t t = msuz,t t

−∫ L

0ϑp((1 + ux,s)uz,ss − uz,sux,ss

),tds

−Cpλ − λ

R= 0 (11)

By spatial integration for ux and utilizing the appro-priate boundary conditions, the Lagrange multiplierbecomes

� = −E Ieffuz,suz,sss − 1

2ms

∫ s

L

(∫ δ

0u2z,ξdξ

),t tdδ

+ (ϑpλ [H(x) − H(x − L)]),suz,s (12)

which can be substituted into the equation for uz to find

⎡⎢⎢⎢⎢⎢⎣

msuz,t t + czuz,t + 12ms

[uz,s

∫ sL

(∫ δ

0 u2z,ξdξ)

,t tdδ

],s

+ E Ieff(uz,sss + uz,s

(uz,suz,ss

),s

),s

− 2γ B(∣∣uz,ss∣∣ uz,sss),s

−ϑpv

[∂δ(x)

∂s− ∂δ(x − L)

∂s

]− ϑpv [δ(x) − δ(x − L)]

(uz,suz,ss

)

⎤⎥⎥⎥⎥⎥⎦

= −msub,t t

Cp v + v

R+∫ L

0ϑp

(uz,ss + 1

2u2z,suz,ss

),tds = 0 (13)

If the base motion is harmonic about the first nat-ural frequency of the structure, then a single modeassumption can be made for primary resonance behav-ior, uz(s, t) = φ(s)η(t), where φ(s) is the mass nor-malized first mode shape of a cantilever,

φ(s) = A1

(cos

λ1s

L− cosh

λ1s

L

+ σ

(sin

λ1s

L− sinh

λ1s

L

))

σ = sin λ1 − sinh λ1

cos λ1 + cosh λ1λ1 = 1.8751∫ L

0msφi (s)φ j (s)ds = δi j (14)

This results in the equations of motion in modalcoordinates,

m∗η + c∗η + k∗ (1 − γ ∗ |η|) η + α∗

L2 η3

+ β∗

L2

(ηη2 + η2η

)− θpv − θNLvη2 = G∗ cos�t

Cp v + v

R+ θpη + θNLη2η = 0 (15)

which can be converted back into the measurementcoordinates by w = w(Lm, t) = φ(Lm)η(t), where

Lm is the distance from the base of the cantilever wherethe velocity is experimentally measured, yielding thefinal equations of motion which account for nonlinearstrain and the inextensibility condition of a cantilever:

mw + cw + k(1 − γ |w|)w + α

L2w3

+ β

L2

(ww2 + w2w

)− θpv − θNL

φ2(Lm)vw2 = G cos�t

Cp v + v

R+ θp

φ(Lm)w + θNL

φ3(Lm)w2w = 0 (16)

where

m = ms∫ L0 φ2(s)ds

φ(Lm)∫ L0 φ(s)ds

,

γ = 2γ B∫ L0 φ(s)

(φ′′(s)φ′′′(s)

)′dsE Ieffφ(Lm)

∫ L0 φ(s)φ′′′′(s)ds

,

c = cz∫ L0 φ2(s)ds

φ(Lm)∫ L0 φ(s)ds

,

α =E Ief f L2

∫ L0 φ(s)

{φ′(s)

(φ′(s)φ′′(s)

)′}′ds

φ3(Lm)∫ L0 φ(s)ds

,

k = E Ieff∫ L0 φ(s)φ′′′′(s)ds

φ(Lm)∫ L0 φ(s)ds

,

β =msL2

∫ L0 φ(s)

[φ′(s)

∫ sL

∫ δ

0 φ′(s)2dξdδ]′ds

φ3(Lm)∫ L0 φ(s)ds

,

θp = ϑpφ′(L), θNL = 1

2ϑp(φ′(L)

)3,

G = msab (17)

where m, c, and k are the equivalent linear mass, damp-ing, and stiffness terms, γ accounts for piezoelectricsoftening, α is the geometric hardening coefficient,

123

1940 D. Tan et al.

β is the inertial softening coefficient, θp is the linearelectromechanical coupling coefficient, θNL is the non-linear electromechanical coupling coefficient resultingfrom nonlinear strain, and G is the forcing term.

The coupled ODEs given by Eq. (16) are solvedusing themethod of harmonic balance [53] for all resis-tive loads and acceleration levels considered in theexperiments. Since the base excitation is assumed tobe harmonic with a driving frequency �, the mechan-ical response solution and voltage output are expectedto have the same period as the mechanical excitationand can be approximated by truncated Fourier seriesexpansions,

wQ =Q∑

q=1

[aq cos(q�t) + bq sin(q�t)

]

vQ =Q∑

q=1

[cq cos(q�t) + dq sin(q�t)

](18)

where Q is the number of harmonics considered and thetypical constant terms a0 and c0 in the Fourier seriesexpansions are zero because of the symmetry of thestructure. Substituting the approximate solutions intoEq. (16) results in the residual functions

R1 =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

mwQ + cwQ + k(1 − γ

∣∣wQ∣∣)wQ + α

L2w3Q

+ β

L2

(wQw2

Q + w2QwQ

)

−(

θp + θNL

φ2(Lm)w2

Q

)vQ − G cos�t

R2 = Cp vQ + vQ

R+ θp

φ(Lm)wQ + θNL

φ3(Lm)w2

QwQ

(19)

Then, using the Galerkin method of mean weightedresiduals,

∫ 2π/�

0R1 cos(q�t)dt=0,

∫ 2π/�

0R2 cos(q�t)dt=0

∫ 2π/�

0R1 sin(q�t)dt=0,

∫ 2π/�

0R2 sin(q�t)dt=0

q = 1, . . . , Q

(20)

a systemof 4Q algebraic equations for the Fourier coef-ficients aq , bq , cq , dq can be generated. Here we chooseto include Q = 5 harmonics, resulting in 20 equationsand 20 unknowns. These equations can be solved usinga multivariate Newton–Raphson method. Since reso-nant excitation of the nonlinear structure results in a

Table 1 Dimensions and material properties of the MFCbimorph

Property Symbol Value

Active length Lact 85 mm

Clamped active length L 75.5 mm

Clamped overall length Lo 83.5 mm

Active width bact 7 mm

Total thickness h 0.61 mm

Measured capacitance Cp 3.40 nF

First natural frequency (SC) fSC 40.5 Hz

saddle node bifurcation, the Fourier coefficients werefirst solved for an off-resonance excitation frequency,and then the driving frequency � was linearly sweptup and down around the first natural frequency usingthe previous solutions for the Fourier coefficients asseed values in the Newton–Raphson method in orderto capture both the high- and low-energy stable solu-tions of the bifurcation. Another method to analyze theoverall nonlinear behavior is by using a multiple scalesapproach [53] on the equation ofmotion in the short cir-cuit condition. This method yields an equivalent nondi-mensional cubic nonlinearity coefficient which com-bines the effect of the geometric hardening from α andthe inertial softening from β:

αeq = α

k− 2

3

β

m(21)

which is expected to be positive for the first bendingmode, indicating that the geometric hardening termdominates the third-order behavior of the system.

3 Experimental setup

Two MFCs, model M8507-P1 from Smart MaterialCorp., were vacuum bonded together without a sub-strate to form a bimorph structure, the dimensions andmaterial properties of which are listed in Table 1. Thesamplewas then placed in an aluminum clamp and ontoan APS 113 Electro-Seis long stroke shaker, as shownin Fig. 2. An accelerometer was mounted to the baseof the clamp using wax, and the velocity of the MFCbimorph was measured near the free end of the can-tilever with a Polytec OFV-505 laser Doppler vibrom-eter. The MFC samples in the bimorph were wired in

123

Resonant nonlinearities of piezoelectric macro-fiber 1941

Fig. 2 Cantilevered MFC bimorph mounted on a long strokeshaker for base excitation and shunted to a resistive electricalload box

parallel and placed in the short circuit condition. Lin-ear noise burst experiments were performed to identifythe fundamental short circuit (SC) resonance frequency(which is approximately the SC natural frequency dueto low damping) and then repeated for resistive loadsof 1 k�, 1.3 M�, and 100 M�.

Using a VCS 201 vibration controller, the root-mean-square (RMS) acceleration level was set to 0.5gand an up and down frequency sweep was performedaround the first short circuit natural frequency at1 Hz/min. The frequency sweep was repeated at 0.4g,0.3g, 0.2g, and 0.1g RMS, for the cases of resistiveloads of a short circuit, 1 k�, 1.3 M� and 100 M�.The base acceleration level, tip velocity of the MFCbimorph, as well as the voltage across the resistiveload were measured using an NI-9223 data acquisitiondevice during the nonlinear frequency sweeps.

4 Results

4.1 Experimental results

Figure 3 shows the mechanical and electrical responseof the MFC bimorph under base excitation under vari-ous resistive loads including only the data correspond-ing to the downward frequency sweep for clarity. For allacceleration levels and resistive loads considered, thequadratic piezoelectric softening dominates the nonlin-ear behavior (resulting in an initially linear backbonecurve bent to the left), but when the response amplitudeis high enough, the third-order geometric hardeningbehavior becomes visible as expected for the first mode

of an inextensible cantilever (resulting in a quadraticbend of the backbone curve back to the right).

Using the RMS velocity data under the short circuitcondition, one can plot the detuning of the excitationfrequency relative to the short circuit natural frequencyversus the response amplitude inmeters and then createa backbone curve by applying a polynomial fit over thepoints corresponding to the maximum response ampli-tudes, as shown in Fig. 4. This polynomial is of the formσ = C1a2 + C2a + C3, where σ is the detuning anda is the response amplitude. The nonlinear quadraticpiezoelectric softening coefficient γ and the overallnondimensional cubic nonlinearity coefficient αeq canbe approximated from this polynomial by:

γ =1 −

(1 − σ1

ωn

)

a1

αeq = 8C1L2

3ωn(22)

where a1 is a small response amplitude and σ1 is thedetuning corresponding to that small response ampli-tude, which results in γ = 155.32 1

m and αeq = 98.75for this sample.

4.2 Theoretical results

Using the measured capacitance of the MFC bimorph,the effective area Ae was calculated to be 0.0446 mm2.The coefficients found in Eq. (16) were then calcu-lated using Eqs. (3), (5), and (17), as well as withthe parameters extracted from the experimental resultsusingEq. (22). The nonlinear equations ofmotion givenby Eq. (16) were then solved using the method of har-monic balance for all resistive loads and accelerationlevels considered in the experiments as explained pre-viously. The calculated velocity response and outputvoltage are plotted in Fig. 5.

Both the experimental and theoretical RMS voltageplots for the optimal resistive load (1.3 M�) were con-

verted into the average power plot using Pavg = V 2RMS

Rfor all base acceleration levels considered, as shown inFig. 6. The theoretical model shows good agreementwith the experiments performed, accurately predictingthe maximum power output as well as the jump fre-quencies. However, the theoretical model slightly over-predicts the output voltage and subsequently underpre-dicts themechanical response of theMFCbimorph can-

123

1942 D. Tan et al.

Fig. 3 Measured RMS tipvelocity and output voltageof the MFC bimorph versusresistive load and frequencyat RMS base accelerationlevels of 0.1g, 0.2g, 0.3g,0.4g, and 0.5g using adownward frequency sweep(increasing baseacceleration level fromblack to red). (Color figureonline)

Fig. 4 Experimentallydetermined responseamplitude under the shortcircuit condition versusdetuning relative to the shortcircuit natural frequencyand the correspondingquadratic backbone curve.(Color figure online)

Fig. 5 Theoretical RMS tipvelocity and output voltageof the MFC bimorph versusresistive load and frequencyat RMS base accelerationlevels of 0.1g, 0.2g, 0.3g,0.4g, and 0.5g using adownward frequency sweep(increasing baseacceleration level fromblack to blue). (Color figureonline)

tilever. This discrepancy is likely due to other dissipa-tive terms in the electrical domain (e.g., dielectric loss)not accounted for in the present work.

The nonlinear governing equations given byEq. (16)were linearized by setting the nonlinear coefficients γ ,α, β, and θNL to zero. The theoretical average power forthe five acceleration levels considered was calculatedfor various resistive loads at 37, 38, 39, and 40Hzutiliz-ing both the linearized model and the nonlinear model

presented in this work, and the results are comparedwith the experimental results in Fig. 7. As expected, thelinearized model shows the highest output power nearthe linear resonance frequency of the structure (around40.5 Hz) and decreased output power as the drivingfrequency moves away from the linear resonance. Thepredictions of the linearized model are highly inaccu-rate for the realistic base acceleration levels consideredin this work. The experimental data shown in Fig. 3

123

Resonant nonlinearities of piezoelectric macro-fiber 1943

Fig. 6 Experimental (redcircles) and theoretical (bluelines) average power atRMS base accelerationlevels of 0.1g, 0.2g, 0.3g,0.4g, and 0.5g using adownward frequency sweepwith a resistive load of1.3 M�. (Color figureonline)

Fig. 7 Theoretical average power utilizing a linearized model(black dashed lines) and the fully nonlinear model (blue solidlines), as well as experimentally determined average power (redmarkers) at RMS base acceleration levels of 0.1g, 0.2g, 0.3g,

0.4g, and 0.5g using a downward frequency sweep for variousresistive loads sampled at 37, 38, 39, and 40 Hz. (Color figureonline)

exhibit strong quadratic piezoelectric softening behav-ior, resulting in larger response amplitudes at frequen-cies below the short circuit natural frequency whichthe fully nonlinear model accurately predicts. As thebase acceleration level increases, this nonlinear behav-ior becomes more pronounced, leading to drasticallyincreased errors in the linearized model.

5 Conclusions

A nonlinear nonconservative electroelastic modelingframework was developed for energy harvesting appli-cations from base excitation of the 33-mode of anMFCbimorph cantilever around its first bending mode. Theequivalent electromechanical properties of the MFC

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bimorph were calculated using a mixing rules formu-lation, and the effective area of a single RVE as wellas piezoelectric softening coefficient were extractedexperimentally from the short circuit base excitationcase. The equations were solved for primary reso-nance behavior using the method of harmonic balancefor multiple base acceleration levels, and the resultswere experimentally validated over a range of resis-tive electrical loads and excitation frequencies. Theexperimentally observed nonlinear dynamics, whichincludes quadratic material softening, cubic geomet-ric hardening, cubic inertial softening, and a nonlinearelectromechanical coupling term resulting from geo-metric effects, were all accommodated by the nonlinearmodel. It was shown that the linearized model yieldshighly inaccurate results for the realistic base acceler-ation levels considered in this work for typical ambientvibration frequencies, while the nonlinear model pro-posed in this work yields very accurate representationand prediction of the governing dynamics. This frame-work successfully captures the dynamics ofMFC struc-tures for energy harvesting and sensing applicationswith relatively weak electric fields. For applicationssuch as resonant actuation with higher electric fields,additional terms (e.g., coupling nonlinearities, electricfield nonlinearities, dielectric losses) may be requiredand can be incorporated in a similar procedure.

Acknowledgements This work was supported in part by theNSF Grant CMMI-1254262, which is gratefully acknowledged.

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