Original Article

Journal of Intelligent Material Systemsand Structures2018, Vol. 29(4) 514–529� The Author(s) 2017Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1045389X17711788journals.sagepub.com/home/jim

Investigation of direct current powerdelivery from nonlinear vibrationenergy harvesters under combinedharmonic and stochastic excitations

Quanqi Dai and Ryan L Harne

AbstractLeveraging smooth nonlinearities in vibration energy harvesters has been shown to improve the potential for kineticenergy capture from the environment as a transduced, alternating flow of electrical current. While researchers haveclosely examined the direct current power delivery performance of linear energy harvesters, there is a clear need toquantify the direct current power provided by nonlinear harvester platforms, in particular those platforms having bis-table nonlinearities that are shown to have advantages over other smooth nonlinearities. In addition, because real worldexcitations are neither purely harmonic nor purely stochastic, the influences of an arbitrary combination of such excita-tion mechanisms on power delivery must be uncovered. To bring needed light to these roles and opportunities for non-linear energy harvesters to provide direct current electrical power for numerous applications, this research formulates anew analytical approach to characterize simultaneous harmonic and stochastic mechanical and electrical responses ofnonlinear harvester platforms subjected to realistic base excitation. Based on the outcomes of analytical, numerical, andexperimental studies, it is found that additive stochastic excitation may result in direct current power enhancement viaperturbation from a low amplitude state particularly at low frequencies or reduce the direct current power by prevent-ing persistent snap-through response often at higher frequencies. When the noise standard deviation is greater than theharmonic amplitude by approximately two times, the advantages to direct current power generation are more oftenrealized.

KeywordsEnergy harvesting, nonlinear dynamics, direct current power

Introduction

Traditionally, the maintenance of engineered infra-structures is undertaken according to a scheduled basis,which is ineffective in the long-term due to accumu-lated expense and the serious potential for unintendedfailures (Randall, 2011). Condition-based structuralhealth monitoring has risen as a promising alternativewhereby the health of the infrastructures is assessed inreal-time through a network of sensor nodes (Randall,2011). Such sensors individually demand from around100 mW to about 100 mW of direct current (DC)power supply, based on the function involved (Anon,2015; Baert et al., 2006; Roundy et al., 2003). Becausemany of the sensors are placed so as to monitor thestructural vibration levels as dynamic health indicators,kinetic energy is available to the sensor nodes. Thus,harvesting the kinetic energy using electromechanicaloscillators collocated with the sensors has emerged as a

way to realize self-sufficient structural health monitor-ing systems (Erturk and Inman, 2011b; Priya andInman, 2009). In order to effectively harvest the kineticenergy from the structural oscillation, the vibrationenergy harvester must be sensitive to the ambientenergy forms, because conventional electromechanicaltransduction mechanisms produce current flow in pro-portion to displacement or velocity of the mechanicalresponse (Erturk and Inman, 2011b; Priya and Inman,2009). Numerous researchers have found that thesmooth nonlinearities of monostable and bistable

Department of Mechanical and Aerospace Engineering, The Ohio State

University, Columbus, OH, USA

Corresponding author:

Ryan L Harne, Department of Mechanical and Aerospace Engineering,

The Ohio State University, Columbus, OH 43210, USA.

Email: harne.3@osu.edu

Duffing oscillators may lead to broader spectral sensi-tivities to ambient vibrations than linear resonators,which has motivated considerable research attention toscrutinize the opportunities (Leadenham and Erturk,2015; Mann and Sims, 2009; Masana and Daqaq,2011a, 2011b; Tang et al., 2010).

The non-resonant nature of snap-through dynamicsof bistable energy harvesters is a particularly advanta-geous feature for transduction, since the unique, largeamplitude motions may be tuned and triggered withrelatively low level input vibrations across a broadrange of frequencies (Cottone et al., 2009; Daqaq,2011; Erturk et al., 2009; Harne and Wang, 2014b,2017; Stanton et al., 2010; Virgin, 2000). Despite thestrong nonlinearity involved which challenges manytheoretical tools, several research teams have studiedbase excited bistable energy harvesters coupled withresistive electrical circuits through rigorous analyticalmethods to assess alternating current (AC) power deliv-ery opportunities. For instance, Stanton et al. (2012),Panyam et al. (2014), Daqaq (2011), and Harne andWang (2014b) have leveraged various theoreticalapproaches to explicitly predict the AC power genera-tion from bistable vibration energy harvesters whensubject to either harmonic or stochastic base excita-tions. Despite the value of such insights, in order tosupply useful power for structural monitoring sensornodes, DC power must be delivered from the harvester.Thus, a rectifier circuit is needed to interface betweenthe oscillatory electromechanical response and the elec-trical load. Rectifier circuits are dynamic systems, arenonlinear, and coupled to the dynamic response of theharvester itself, thus demanding special attention inmodel approaches (Liang and Liao, 2012; Shu andLien, 2006a). Researchers have found that active syn-chronized switch harvesting on inductor circuits(Guyomar et al., 2005; Liang and Liao, 2012; Shuet al., 2007) and synchronized electric charge extractioncircuits (Badel and Lefeuvre, 2016; Lefeuvre et al.,2005; Wu et al., 2012) enable promising DC powerdelivery for linear vibration energy harvesters underharmonic excitations. Comparatively, the simple andpassive standard diode bridge rectifier provides aneffective DC power delivery from linear and nonlinearvibration energy harvesters (Elvin, 2014; Pasharaveshet al., 2017; Shu and Lien, 2006b; Sodano et al., 2005)without unfavorable loss of power resulting from activesynchronization. For instance, Shu and Lien (2006b)have shown that the electromechanical coupling coeffi-cient is strongly influential on the mechanical-to-electrical energy conversion efficiency of linear energyharvesters interfaced with diode bridge rectifiers, whichemphasizes the importance of comprehensively investi-gating the roles of practical rectifying circuits in con-trast to AC resistive circuit counterparts. Yet, despitethe significance of this knowledge, the insights are rele-vant only for linear vibration energy harvesters, while

the analytical methodologies are not directly extensibleto the numerous cases in which energy harvesters pos-sess nonlinearities for performance enhancement.

Yet, considering the broad range of knowledgerevealed by the studies surveyed above, such insightspertain only to the responses of energy harvesters whensubjected to pure harmonic or pure stochastic excita-tions. In fact, numerous environments in which energyharvesters will be deployed alongside structural moni-toring sensors do not exhibit such simple purely harmo-nic or stochastic excitation spectra. Instead, real worldexcitations contain a combination of noise and periodiccontributions, which is inevitable based on the fact thatthe monitored structures oscillate in primary modesbut are themselves acted upon by random excitationssuch as tire–road interaction or footfalls (Green et al.,2013; Turner and Pretlove, 1988; Zuo and Zhang,2013). There is a clear lack of investigations on energyharvester electrodynamic responses that result fromexcitation conditions more closely representative of rea-listic vibrations. Although the dynamic sensitivities ofweakly nonlinear oscillators have been examined undersuch conditions (Anh and Hieu, 2012; Bulsara et al.,1982; Nayfeh and Serhan, 1990), and a recent analysishas been presented to explore the purely mechanicalresponses of post-buckled structures to combined har-monic and stochastic perturbations (Harne and Dai,2017), no such analysis exists to characterize the elec-tromechanical responses and DC power delivery ofnonlinear vibration energy harvesters subjected to arbi-trary combinations of harmonic and stochastic baseaccelerations.

Motivated to close this fundamental knowledge gap,this research undertakes comprehensive efforts to char-acterize the DC power delivery from nonlinear vibra-tion energy harvesters subjected to a combination ofharmonic and stochastic excitation. Due to the strategicadvantages of bistable nonlinearities, attention isdirected to bistable energy harvesters, although themodel and analytical methods presented pertain to thebroader class of smooth, stiffness-based nonlinearities.The comprehensive efforts undertaken here notablyinclude a new analytical procedure that facilitates directprediction of the structural dynamics and electricalresponses (including DC power), in parallel with atrack of numerical simulation for verification andexperiments for validation of the theoretical approach.The following sections first introduce the nonlinear bis-table vibration energy harvester platform examinedand describe the corresponding analytical frameworkestablished to investigate the platform. Then, a comple-ment of analytical, numerical, and experimental resultsare presented to uncover the influences of the combinedexcitation form and electrical circuit parameters on theDC power delivery. A summary of primary conclusionsand directions for future research are provided in thefinal section.

Dai and Harne 515

Nonlinear vibration energy harvestermodeling

Energy harvester platform

This section introduces the experimental platformaround which the model formulation is established, whilefurther modeling details and analytical procedures aregiven in section ‘‘Governing equations,’’ ‘‘Approximateanalytical solution to governing equations,’’ ‘‘Harmonicelectromechanical dynamics,’’ ‘‘Stochastic electromecha-nical dynamics,’’ and ‘‘Combined harmonic and stochas-tic response of the nonlinear energy harvester.’’ In thisstudy, the energy harvester under consideration employsa pair of attractive magnets that act on a cantileveredpiezoelectric energy harvester. The positioning of themagnets with respect to the ferromagnetic cantilever tiptailors the type and strength of the nonlinearity (Erturkand Inman, 2011a; Feeny and Yuan, 2001; Hikihara andKawagoshi, 1996; Moon and Holmes, 1979). Figure 1provides a photograph and schematic of the experimen-tal system examined in the research. A piezoelectric beam(PPA-2014; Mide Technology) is used as the energy har-vester structure. The device is composed of layers ofPZT-5H, copper, and glass-reinforced epoxy FR4; thiscomposition results in appreciable inherent mechanicaldamping due to both the laminated design and the FR4in particular. This beam is clamped at one end of a rigidaluminum mount. Attached to the free end of the cantile-ver are steel extensions, with total mass m = 9 g, thatreduce the lowest order natural frequency of the beamand provide for a ferromagnetic material for the magnetsto act upon. The total length of the cantilever and exten-sion is L = 61 mm. A pair of neodymium magnets ispositioned near the ferromagnetic extension, which causeattractive forces that work in opposite directions to lin-ear elastic forces induced in the beam by displacement ofthe beam tip (Moon and Holmes, 1979). By reducing thedistance between the beam tip and magnets, d, and thedistance between the magnets, D, the energy harvestermay become bistable, thus maintaining one of two stati-cally stable equilibria. In this work, the bistable nonli-nearity is focused upon in the investigations, althoughthe model framework composed for study is extensibleto the significantly greater class of smooth, stiffness-based nonlinearities. As shown in Figure 1(b), the electri-cal leads from the energy harvester are connected to astandard diode bridge (1N4148 diodes). After the bridge,the DC flows through a parallel arrangement of filteringcapacitor Cr and resistive load R, across which the recti-fied voltage, and DC power, are evaluated.

Governing equations

Considering that the nonlinear energy harvester exhi-bits a weak mono- or bistable nonlinearity, has a lowestorder mode at a frequency much less than higher order

modes, and is excited at frequencies near to this lowestorder mode, the governing equations for this systemare shown to be (Harne and Wang, 2014a; Moon andHolmes, 1979; Panyam et al., 2014)

m€x+ c _x+ k1 1� pð Þx+ k3x3 +anp =�m€z ð1aÞ

Cp _vp + I =a _x ð1bÞ

The assumptions described above are all borne outin the ensuing experimentation. In equation (1), x is therelative displacement of the energy harvester tip (i.e.the end of the extension in the experimental platform)with respect to the motion of the base z to which theclamped end of the harvester is attached; m, c, k1, k3

are the contributions from the cantilever mass, viscousdamping, linear, and nonlinear stiffness to lowest ordermode dynamics; p is referred to as a load parameter

Figure 1. (a) Photograph and (b) schematic of experimentalsetup.

516 Journal of Intelligent Material Systems and Structures 29(4)

since it characterizes the significance of magnetic forceinfluence on reducing the linear stiffness similar to anaxial compressive load (Moon and Holmes, 1979); Cp isthe internal capacitance of the harvester; vp is the vol-tage across the piezoelectric beam electrodes; a is theelectromechanical coupling constant; and the overdotoperator denotes differentiation with respect to time t.

The diode bridge AC–DC converter is connected tothe energy harvester as shown in Figure 1(b). Here, per-fect rectification is assumed. I(t) is the AC that flowsinto the diode bridge and is related to the rectified vol-tage vr through (Shu and Lien, 2006a)

I tð Þ=Cr _vr +

vr

R; if vp = vr

�Cr _vr �vr

R; if vp =�vr

0; if vp

�� ��\vr

8>><>>: ð2Þ

To maintain a steady rectified voltage vr for the load,which may be used for supplying a sensor or charging atemporary battery reserve, a sufficiently large capacitorCr is needed (Shu and Lien, 2006a) so that the electricaltime constant RCr is much larger than the period ofmechanical oscillation.

Non-dimensionalization of the governing equationsassists in the acquisition of explicit analytical solutionsvia computationally efficient algorithms. Here, a newnon-dimensionalization approach is introduced due tothe presence of nonlinearities in the energy harvestingstructure and in the rectifying circuit. After non-dimen-sionalization, the governing equations become

x00+hx0+ 1� pð Þx+bx3 + kvp =� z00 ð3aÞ

v0p + I = ux0 ð3bÞ

I tð Þ=gv0r + rvr; if vp = vr

�gv0r � rvr; if vp =�vr

0; if vp

�� ��\vr

8<: ð3cÞ

The corresponding non-dimensional parameters aredefined as follows

t =v0t; v0 =ffiffiffiffiffiffiffiffiffiffik1=m

p; z= z=x0

b= k3x20=k1; h= c=mv0;

k=aV0=k1x0; g =Cr=Cp;

r= 1=CpRv0; u=ax0=CpV0

where x0, V0 are characteristic length and voltage, suchthat x= x=x0, vp = vp=V0, and vr = vr=V0. Here, ( )0

indicates differentiation with respect to non-dimensional time t.

In order to analytically investigate realistic combina-tions of excitations on the nonlinear energy harvesterfor the first time, the base acceleration excitationincludes harmonic and stochastic components

�z00= a cos vt +sw tð Þ ð4Þ

such that w(t) is a Gaussian white noise process with

w tð Þh i= 0 and w tð Þw t + t0ð Þh i= d t0ð Þ ð5Þ

and where s is the normalized standard deviation ofthe noise (Roberts and Spanos, 1990). Considering theharmonic excitation component, a is the normalizedbase excitation magnitude and v is the angular fre-quency of excitation v0 normalized with respect to v0.Thus, the absolute base excitation magnitude and stan-dard deviation are a= ax0k1=m and s =sx0k1=m,respectively.

Approximate analytical solution to governingequations

By leveraging both harmonic and stochastic lineariza-tion, a strategy to predict mechanical responses ofpost-buckled structures was recently introduced (Harneand Dai, 2017). Yet, in the current context of vibrationenergy harvesting, the prior analysis is insufficient sinceit does not account for electrodynamic responses andhas no means of characterizing DC power delivery.This research thus builds significantly beyond the priorwork (Harne and Dai, 2017) for a comprehensive andfirst analysis suitable for nonlinear energy harvesterssubjected to arbitrary harmonic and stochasticexcitations.

To begin, a linearized governing equation system iscreated to approximate the influences of the nonlineari-ties in equation (3). Since the nonlinearity is presentonly in equation (3a), the equivalent linear system isgoverned by

x00+hx0+ve2x+ e+ kvp =�z00 ð6aÞ

v0p + I = ux0 ð6bÞ

I tð Þ=gv0r + rvr; if vp = vr

�gv0r � rvr; if vp =�vr

0; if vp

�� ��\vr

8<: ð6cÞ

The parameter v2e is an equivalent linear natural fre-

quency while e is an equivalent offset of displacementthat exists if the nonlinearity induces bistability.Specifically, for bistable nonlinearities in the energyharvester, e= 0 corresponds to snap-through responsesthat are symmetric about the unstable equilibrium,while e 6¼ 0 correspond to intrawell responses that oscil-late with low displacement amplitude around one of thenon-zero stable equilibria. To determine the relationsbetween these parameters and the original parametersof the exact governing equation system (equation (3)),the mean-square error E between equations (3a) and(6a) is minimized

Dai and Harne 517

E = 1� pð Þx+bx3 � v2ex� e ð7Þ

The mean-square error occurs when ∂ E2� �

=∂v2e = 0

and ∂ E2� �

=∂e= 0 (Roberts and Spanos, 1990). By tak-ing advantage of the linearized governing equations(equation (6)), to derive such relations, the mechanicaland electronical responses are assumed to be composedfrom a pair of contributions strictly associated witheither the harmonic or stochastic excitation compo-nents (equation (4)). A similar concept has been estab-lished to predict mechanical responses of post-buckledoscillators that are subjected to combined harmonicand stochastic excitation (Harne and Dai, 2017). Here,this concept is significantly extended to consider theoverall electrodynamic responses of nonlinear energyharvesters, including DC power delivery, according tothe requirements to account for the intricate electrome-chanical coupling and rectification stages. Using sub-script h to denote the harmonic contribution and r todenote the random or stochastic contribution, one has

x tð Þ= xh tð Þ+ xr tð Þ ð8aÞ

vp tð Þ= vp,h tð Þ+ vp,r tð Þ ð8bÞ

vr tð Þ= vr,h tð Þ+ vr,r tð Þ ð8cÞ

Note that the components associated with the whitenoise have zero-mean, xrh i= vp,r

� �= vr,r

� �= 0. The

harmonic components are written by a Fourier seriesexpansion associated with the normalized frequency ofthe base acceleration, while vr,h(t) is assumed to beconstant

xh tð Þ= k tð Þ+ h tð Þ sin vt + g tð Þ cosvt

= k tð Þ+ n tð Þ cos vt � f tð Þ½ �ð9Þ

vp,h tð Þ= p tð Þ sin vt + q tð Þ cos vt ð10Þ

The coefficients k, h, and g vary slowly in time, whilen2 = h2 + g2 and tan f= h=g. If the energy harvesterexhibits a bistable nonlinearity, the constant coefficientk is required to realize the displacement bias while thepiezoelectric voltage associated with the harmonic baseacceleration does not require such accounting accordingto equation (3b). Then, the mean-square error is mini-mized when

∂ E2� �∂v2

e

= 1� pð Þx2 � v2ex2 +bx4 � ex

� �= 0 ð11aÞ

∂ E2� �∂e

= 1� pð Þx� v2ex+bx3 � e

� �= 0 ð11bÞ

Substitution of the assumed solution equation (8a)into equation (11) enables the determination of v2

e ande (Ibrahim, 1985)

v2e = 1� pð Þ

+3

4b

n4 + 8n2 x2r

� �+ 8 x2

r

� �2+ 4k2 n2 + 2 x2

r

� �� �n2 + 2 x2

r

� �ð12Þ

e=1

4bk

3n4 � 8k2 n2 + 2 x2r

� �� �n2 + 2 x2

r

� � ð13Þ

Because equation (6) is linear, the combinedresponse of the linearized system is the superposition ofresponses individually associated with the harmonic orstochastic excitation components. Thus, the simulta-neous solution to equations (14) and (15) enables thedetermination of the unknowns within the assumedsolution forms of equation (8)

x00h +hx0h +v2exh + e+ kvp,h = a cos vt ð14aÞ

vp,h0+ Ih = uxh

0 ð14bÞ

Ih tð Þ=gvr,h

0+ rvr,h; if vp,h = vr,h

�gvr,h0 � rvr,h; if vp,h =� vr,h

0; if vp,h

�� ��\vr,h

8<: ð14cÞ

x00r +hx0r +v2exr + e+ kvp,r =sw tð Þ ð15aÞ

vp,r0+ Ir = ux0r ð15bÞ

Ir tð Þ=gvr,r

0+ rvr,r; if vp,r = vr,r

�gvr,r0 � rvr,r; if vp,r =�vr,r

0; if vp,r

�� ��\vr,r

8<: ð15cÞ

Harmonic electromechanical dynamics

To predict the harmonic electromechanical dynamicsof the nonlinear energy harvester coupled to the rectify-ing circuit, we build upon a strategy devised by Liangand Liao (2012) that was developed for linear energyharvesting platforms. Introducing a total phasevF=vt � f, the periodic voltage vp,h is presented inFigure 2 as the solid green curve. When the displace-ment xh reaches a peak, the vp,h begins to fall from thelevel of the rectified voltage vr,h until the time at whichthe condition of equation (14c) is met for the negativevalue of the vp,h that matches the absolute value of thesteady-state rectified voltage.

Considering the sinusoidal form of the electromecha-nical responses and the switching conditions of equa-tion (14c), the harmonic voltage across the piezoelectricelectrodes vp,h is expressed

vp,h vFð Þ=

un cosvF� 1½ �+ vr,h; 0\vF�Y�vr,h; Y\vF�p

un cosvF+ 1½ � � vr,h; p\vF�p +Yvr,h; p +Y\vF� 2p

8>><>>:

ð16Þ

As shown in Figure 2, Y corresponds to the phaseangle at which current flow changes from the blocked

518 Journal of Intelligent Material Systems and Structures 29(4)

to passing states. Using equation (16), this phase isfound to be

cos Y= 1� 2vr,h

unð17Þ

according to the continuity of the piecewise functions.To leverage the harmonic steady-state assumptions ofequation (14), a sinusoidal form of equation (16) isrequired. Thus, vp,h is approximated by its fundamentalterms of its Fourier series expansion. Using the factthat for one-half cycle of the steady-state behaviors(Figure 2) the electromechanical responses oscillatefrom peak-to-peak (Shu and Lien, 2006a), one finds

ðp0

vp,h0 � ux0d(vF)=�2vr,hv+ 2unv ð18Þ

In addition, assuming that the rectified voltagechange per cycle is negligible compared to the meanvalue, an energy balance yields

ðp0

gvr,h0+ rvr,hd(vF)=prvr,h ð19Þ

Therefore, integrating equation (14c) in a semi-period yields

vr,h =2u

pv

r+ 2n ð20Þ

Consequently, the DC power associated with theharmonic base acceleration is

Ph =vr,h

2

R=

4u2

R pv

r + 2� �2

n2 ð21Þ

Then, the first-order time-harmonic of vp,F is foundby the fundamental term of the Fourier series of equa-tion (16) using equation (20)

vp,F tð Þ= �g tð Þp

u sin2 Y+h tð Þu

2p2Y� sin 2Yð Þ

� sinvt

+h tð Þ

pu sin2 Y+

g tð Þu2p

2Y� sin 2Yð Þ�

cos vt

ð22Þ

The prior steps to yield equation (22) are compara-ble to the prior work (Liang and Liao, 2012). Yet, toleverage equation (22) for the nonlinear energy harvest-ing system of interest here, continued steps are requiredto provide meaningful predictions of the dynamicresponse. By virtue of the assumption of single harmo-nic response, it is here assumed that vp,F(t)’vp,h(t).Then, using all of the assumed, harmonic electromecha-nical responses, the coefficients of the constant, sin vt,cos vt terms are collected from substitution of theresponse forms into equation (14a)

�hk0=ve2k + e ð23aÞ

�hh0+ 2vg0=Lh�Xg ð23bÞ

�2vh0 � hg0=Lg +Xh� a ð23cÞ

where

L= ve2 � v2

� �+

uk

2p2Y� sin 2Yð Þ; X=hv+

uk

psin2 Y

ð24Þ

Under steady-state conditions such that the time var-iations on the left-hand side of equation (23) vanish,one determines that

k2 = 0 or k2 =e2

v4e

=� 3

2n2 � 3 x2

r

� �� 1� pð Þ=b ð25Þ

Then, equations (23b) and (23c) together yield

½L2 +X2�n2 = a2 ð26Þ

Equation (26) is a cubic polynomial in terms of n2.The complex roots and negative real roots of equation(26) are therefore not physically meaningful.

Stochastic electromechanical dynamics

Equation (26) characterizes the harmonic displacementresponse of the nonlinear energy harvester, out of whichthe corresponding piezoelectric beam voltage and recti-fied voltage are, respectively, computed from equations(22) and (20). Yet, it is seen that the polynomial ofequation (26) is a function of the equivalent linear natu-ral frequency ve which itself is a function of the mean-square displacement xrh i associated with the stochastic

Figure 2. Representation of single period of voltage across thepiezoelectric and displacement of the energy harvester, asadapted from Liang and Liao (2012).

Dai and Harne 519

excitation component. Therefore, by the linearizationapproach to equation (6), the harmonic and stochasticcontributions become intertwined in the analysis. As aresult, with solutions to the harmonic-related equationsystem (equation (14)), the stochastic-related equationsystem (equation (15)) is addressed.

To determine relations among the mean-square non-linear harvester displacement xrh i and the piezoelectricvp,r and rectified vr,r voltages, a new empirical approachis undertaken. Specifically, a large number of simula-tions are evaluated using a range of non-dimensionalparameters r and g where a non-dimensional beamvelocity x0r is prescribed using zero-mean white noise.Considering the assumption of linearity, one has that(x2

r )0

D E’ x2

r

� �v2

e . Fitting the results with surfaces amongr, g, and xrh i, the following relationships are revealed

vp,2r

� �’

200 x2r

� �ve

2

3r2 + 6r + g1:5

2000

ð27Þ

vr,2r

� �’

ffiffiffi2p

vp,2r

� �ð28Þ

With equations (27) and (28), of the system equation(15) only equation (15a) requires solution. Yet, thisequation is not amenable to straightforward solution.A second linear approximation is then made, assuminga new equivalent natural frequency ven, which assiststo approximate equation (15a) by

x00r +hx0r +ven2xr =sw(t) ð29Þ

The error between equations (15a) and (29) is

En =ve2xr + kvp,r � ven

2xr ð30Þ

Using the same procedures as in the prior section,one has

∂ En2

� �∂v2

en

=∂ ve

4xr2 � 2ve

2ven2xr

2 +ven4xr

2 + 2k(ve2 � ven

2)vp,rxr + vp,r2k2

� �∂v2

en

= 0 ð31Þ

Toward solving equation (31), the expectation of theproduct of xr and vp, r

is taken according to the covar-iance process (Soong, 2004)

E xrvp,r

�=std xrð Þstd vp,r

� �=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi200

3r2 + 6r + g1:5

2000

sx2

r

� �ve

ð32Þ

Consequently, the new equivalent natural frequencyis

ven2 =ve

2 + kve

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi200

3r2 + 6r + g1:5

2000

sð33Þ

Thereafter, the mean-square displacement xrh iinduced by the zero-mean, white noise base excitationcomponent is (Bulsara et al., 1982)

x2r

� �=

s2

2hv2en

ð34Þ

With equation (34), the piezoelectric and rectifiedvoltages associated with the stochastic excitation arefound from equations (27) and (28), respectively.

Combined harmonic and stochastic response of thenonlinear energy harvester

Considering the prior analytical formulation and non-linear coupling of equations, equations (12), (26), and(34) must be simultaneously solved to compute the totalresponse of the energy harvester when subjected to acombination of harmonic and stochastic base accelera-tion. Then, by equation (8), the total responses may bereconstructed.

In addition to the response forms reconstructed inequation (8), specific measures are useful to character-ize the energy harvesting performance of the nonlinearplatform. For instance, one meaningful measure is thetotal mean-square displacement

x2� �

= x2h

� �+ x2

r

� �= k2 +

1

2n2 + x2

r

� �ð35Þ

The total mean-square voltage across the piezoelec-tric electrodes is given by equation (36). Finally, thetotal mean-square rectified voltage across the resistiveload R is computed from equation (37) which leads tothe determination of the total DC power delivery to theload via equation (38)

vp2

� �=

1

2vp,h

�� ��2 + vp,2r

� �ð36Þ

vr2

� �= vr,h

2 + vr,2r

� �ð37Þ

P=Ph +vr,r

2� �

Rð38Þ

All together, the new analytical approach buildsgreatly from prior work that studies the harmonicallyexcited response of linear energy harvesters coupledwith rectifying circuits (Liang and Liao, 2012) andfrom prior work that studies the structural dynamics ofpost-buckled oscillators subjected to harmonic and sto-chastic excitations (Harne and Dai, 2017). Indeed, theanalysis of this report enables the first approach to

520 Journal of Intelligent Material Systems and Structures 29(4)

explicitly predict the structural and electrical dynamicsof nonlinear vibration energy harvesters subjected toarbitrary combinations of harmonic and stochasticexcitations when coupled to rectifying circuit for DCpower delivery.

Experimental methods

The experimental platform is shown in Figure 1(a).Laser displacement sensors (Micro Epsilon ILD-1420)measure the absolute cantilever tip displacement andabsolute displacement of the electrodynamic shakertable. An accelerometer (PCB Piezotronics 333B40)also measures the shaker table acceleration in the sameaxis of motion. The base excitations are applied by acontrolled electrodynamic shaker (LabWorks ET-140),as driven by an amplifier (LabWorks PA-141). Thenoise excitation component is defined as the variance ofa normally distributed voltage time series that superim-poses with a harmonic voltage time series, whichtogether drive the shaker. The AC piezoelectric voltageacross the electrodes of the energy harvester is recordedin addition to the rectified voltage evaluated over theresistive load. All channels of data are recorded at asampling frequency of 4096 Hz and are thereafter digi-tally filtered from 1 to 500 Hz. The experimentally iden-tified parameters of the nonlinear energy harvester anddiode bridge circuit are provided in Table 1. As seen inthe table, the load parameter p is greater than unity,which indicates that the energy harvester was subjectedto enough magnetic force to induce bistability.

Results and discussions

In order to verify the analytical predictions, directnumerical simulations of the governing equations arecarried out using fourth-order Runge–Kutta numericalintegration with increased tolerances on adaptive time-stepping routines in the MATLAB software. Initialconditions for the states of the electromechanical sys-tem are chosen from normally distributed randomnumbers approximately within an order of magnitudeof the ultimate response state amplitude. The analyticaland numerical efforts are undertaken in parallel withexperimentation, using the parameters identified fromthe experimental platform, as given in Table 1. Todetermine the natural frequencies associated with eachlow amplitude intrawell response and the damping con-stant, impulsive ring down responses are undertaken. Itis ensured that the influences of the magnets upon theferromagnetic beam tip are positioned such that thenatural frequencies around both stable equilibria areidentical to v0, thus leading to only parameters k1 andk3. The distance between statically stable equilibria ismeasured, 2x�. Using classical relations, the equivalentmass of the harvester m is determined by considering itto be a cantilever with tip mass, while the linear

stiffness k1 is similarly identified according to the piezo-electric beam properties and dimensions. With thisinformation, the load parameter is computed asp= 1+v2

0m=2k1 for x� 6¼ 0 and as p= 1� v20m=k1 for

x�= 0. Finally, the nonlinear stiffness k3 is determinedfrom k3 = k1(p� 1)=(x�)2 for x� 6¼ 0 which is the rele-vant case of this research that gives attention to the bis-table nonlinearity of the energy harvester. Formonostable energy harvesters, the determination of thenonlinear stiffness k3 can be achieved using data fromforced excitation experiments (Rao, 2004).

The following paragraphs present the collective ana-lytical, numerical, and experimental results that enablethe detailed study of excitation and electrical parameterinfluences upon the nonlinear energy harvester DCpower delivery.

Electromechanical responses induced by pureharmonic excitation

As a limiting case of the model predictions, the vibra-tion energy harvester is first subjected to base accelera-tion of amplitude a= 7:5m=s2 without noisecontribution s= 0. The linearized natural frequency ofthe bistable nonlinear energy harvester is around25.36 Hz, and the frequencies of the harmonic excita-tion are applied around this resonant state.

For this excitation condition, Figure 3(a) and (b)presents the analytical (curves) and numerical (datapoints) results of the beam tip harmonic displacementamplitude and average charging power, respectively,while the corresponding experimental measurementsare shown in Figure 3(c) and (d). According to the ana-lytical and simulation results, for harmonic excitationfrequencies less than around 20 Hz, the bistable energyharvester may exhibit either the large amplitude snap-through dynamics or the low amplitude ‘‘intrawell’’responses associated with oscillations around one orthe other stable equilibria. Such coexistence of dynamicregimes is revealed by more than one analytical predic-tion or numerical simulation data point for the sameharmonic excitation frequency less than about 20 Hz,in Figure 3(a) and (b). At higher frequencies, only thelow amplitude oscillation is found. Together, the analy-tical and numerical results are in overall good agree-ment across the whole frequency range, excepting

Table 1. Experimentally identified system parameters.

m (g) b (N s/m) k1 (N/m) p (dim)

9.45 0.125 160 1.75

k3 (MN/m3) Cp (nF) Cr (mF) a (mN/V)

33 96 10 1.4

Dai and Harne 521

around 20–25 Hz where the simulations suggest aperio-dic or chaotic states of vibration may occur, which theanalysis is unable to predict by virtue of steady-stateassumptions. Considering these different dynamicregimes, it is clear from Figure 3(b) that the snap-through dynamics are the favored behavior for enhan-cing DC power delivery. Indeed, in Figure 3, a range ofload resistances is considered: R= ½4:7 , 47, 470� kO.By change of the resistance from 4.7 to 47 kO and thento 470 kO, the activation of the snap-through responseis suppressed across portions of the harmonic excita-tion frequency bandwidth. More importantly, consider-ing this large range of resistance, an optimal value issuggested. Namely, the peak DC power generationincreases from around 0.62 mW using a resistance of4.7 kO to around 2.88 mW using 47 kO; then byincreasing the resistance still further to 470 kO, thepeak power reduces to just 2.70 mW. These trends arein good agreement between the simulations and theore-tical predictions, providing strong verification to theanalytical formulation created here.

In the corresponding series of experiments, harmo-nic base acceleration is applied with slowly sweepingfrequency from low to high and then high to low val-ues at a rate of 0.09 Hz/s. The experimental resultsshown in Figure 3(c) and (d) are in good qualitative

and quantitative agreement with the model findings.In particular, the potential for an optimal resistancefor DC power delivery is also apparent in the measure-ments since the resistor of 47 kO provides the highestpeak DC power among the three values of resistancesexamined. Also, the bandwidth of the snap-throughdynamic regime is suppressed by increase in the resis-tance, which is likewise revealed both analytically andnumerically.

Overall, the results of Figure 3 establish that the ana-lytical model accurately reproduces the electrodynamicbehaviors of the bistable nonlinear energy harvesterunder the limiting case of pure harmonic excitation.The following section examines the influence of theadditive stochastic excitation contribution on thedynamic response and DC power generation.

Nonlinear energy harvesting under combinedharmonic and stochastic excitations: case example

Then, in addition to the harmonic amplitude of baseacceleration a= 7:5m=s2, stochastic noise is introducedwith standard deviation s= 11:25m=s2 to result in theoverall combined excitation form that acts on the non-linear harvester. In other words, the ratio of noise stan-dard deviation to harmonic amplitude is approximately

Figure 3. Analytical prediction (in lines) and simulated results (in shapes) of (a) beam tip harmonic displacement magnitude at theharmonic excitation frequency and (b) average charging power across resistive loads R, under pure harmonic excitation withamplitude a= 7:5 m=s2. Corresponding experimental measurements are shown in (c, d).

522 Journal of Intelligent Material Systems and Structures 29(4)

3/2. To provide an easier visualization of the differentcases of excitation, the following results only considerthe case when the load resistance is R= 47kO while theexcitations are either purely harmonic or include thenoise to harmonic excitation ratio of 3/2.

In Figure 4(a) and (b), the analytical results areshown for the pure harmonic case (dashed-dot curves)and for the case with the additive noise (solid curves).Figure 4(a) and (b) shows the beam tip harmonic dis-placement amplitude and the average charging power,respectively. As shown in Figure 4(a) and (b), thenumerical results are shown as gray data points for allof the simulations and as filled red squares for theaverages of the simulations at any specific harmonicexcitation frequency. Figure 4(c) and (d) presents thecorresponding experimental findings, with the pureharmonic data sets indicated with solid curves and theshort-time averages of the measurements under thecombined harmonic and stochastic base excitationsplotted as red filled circles.

According to the analytical predictions shown inFigure 4, due to the additive noise excitation

component, the nonlinear harvester loses the ability toundergo coexistent responses, since only one dynamicregime is predicted analytically at any given frequency,across the bandwidth considered. This result is inagreement with the outcomes of the numerical simula-tions. The introduction of the stochastic excitationcomponent causes the displacement amplitudes inFigure 4(a) to reduce considerably from the levels asso-ciated with harmonic snap-through, although the dis-placements are still greater than the low amplitudedynamic regime. The average charging powers shownin Figure 4(b) are correspondingly reduced; at frequen-cies less than about 20 Hz, the powers are approxi-mately one-half of the levels achieved for the harvesterwhen snap-through dynamics are triggered under pureharmonic excitation. Yet, the experimental measure-ments in Figure 4(c) do not discover as great of reduc-tion in the response as predicted analytically andobserved numerically. In particular, while the noise alsosuppresses the coexistence of the dynamic regimes, itdoes not as greatly impact the response amplitudes,whether considering the displacement or charging

Figure 4. Analytical predictions and numerical simulations of (a) beam tip harmonic displacement magnitude at the harmonicexcitation frequency, (b) average charging power across resistive loads R = 47 kO, with harmonic excitation amplitude a= 7:5 m=s2.Dash-dot curves indicate responses without stochastic excitation, s= 0. Thick solid curves indicate response with noises= 11:25 m=s2. Small squares and filled large squares are, respectively, the individual and means results of numerical simulationsconducted at each excitation frequency. Corresponding experimental measurements are shown in (c, d). Solid curves indicateresponses without stochastic excitation, s= 0, while filled circles indicate response with noise s= 11:25 m=s2.

Dai and Harne 523

power. One explanation for this difference betweenexperiment and model is that the overall dissipation ofenergy in the harvester is modeled by a viscous damp-ing. Yet, the piezoelectric beam used experimentallycontains a significant proportion of glass-reinforcedepoxy laminate, which is a viscoelastic material and assuch leads to frequency- and rate-dependent dampingproperties (Fosdick et al., 1998; Ketema, 1998). Thus,the omission of viscoelastic damping in the currentmodel may explain the difference between experimentalmeasurements and model predictions, which encouragesa more accurate model formulation in future to applysuch a nonlinear energy harvester in practice. Despitesuch minor differences in the amplitudes of the displace-ment and DC comparing the model predictions andmeasurements, the experimental results are still in over-all good agreement with analytical prediction as well asthe numerical simulation. Thus, the theoretical formula-tion and solution procedure established here are suffi-ciently validated.

Considering the influence of the stochastic contribu-tion to the overall base excitation upon the harvester,Figure 4(b) and (d), while the peak DC power deliverymay be reduced, a favorable outcome is the eliminationof the coexistence of low amplitude and snap-throughdynamics at low frequencies. This results in a form ofinsurance to obtain a meaningful DC power level closerin value to the result achieved under pure harmonicexcitation. Both model results and experimental mea-surements reveal such behaviors.

To look deeply at the impact of additive stochasticexcitation, experimental time series at 13 Hz are pre-sented in Figure 5. The time series of base accelerationis shown in Figure 5(a) with and without the stochasticcomponent, recalling that the ratio of noise standarddeviation to harmonic amplitude is 3/2, which explainsthe considerable differences between the two time seriesof base acceleration. The beam tip displacement timeseries is shown in Figure 5(b) and (d) while the corre-sponding voltages across the piezoelectric beam electro-des and across the load resistor R = 47 kO are shownin Figure 5(c) and (e). For the low amplitude intrawelloscillation, Figure 5(b) and (c), the additive stochasticexcitation provides a beneficial perturbation away fromthe low amplitude dynamic state, inducing greatermechanical response and hence rectified voltage. This isseen in Figure 5(b) and (c) comparing the time seriesresponse without noise (dashed or dotted curves) tothose measures with noise (solid or dash-dot curves).The corresponding impact on the electromechanicalsnap-through dynamics are shown in Figure 5(d) and(e). According to Figure 5(d) and (e), the additionalstochastic excitation slightly reduces peak-to-peakpiezoelectric voltage generation, although the displace-ment amplitude is not reduced overall. Yet, because thesnap-through response is perturbed from a steady state,the measurements show that the loss of rectified voltage

results in an overall reduction in the charged voltageacross the resistor, and hence less DC power.

DC power delivery for varied levels of stochasticexcitation

As found from the results of the previous section, theadditive stochastic excitation component may signifi-cantly change the electromechanical responses of thebistable nonlinear energy harvester. And as exemplifiedin the time series of Figure 5, the effects of the noise onthe overall behavior are unique whether the underlyingsteady-state response is a coexistent low amplitudeintrawell oscillation or the high amplitude snap-throughdynamic. To closely investigate these influences, thissection examines a wide range of stochastic excitationlevels with respect to a fixed harmonic level of baseexcitation.

Figure 6 presents analytical, numerical, and experi-mental results obtained when the 13 Hz harmonic exci-tation amplitude is a= 2:85m=s2 while the ratio of thenoise standard deviation to harmonic amplitude, s=a,varies from 0 to 4.5. The value for the load resistor forthese results is R = 470 kO. The analytical predictionsare shown by blue solid (red dashed) curves in Figure6(a) and (b) according to the snap-through (intrawell)responses for the harmonic displacement amplitudeand charging power, respectively. Numerical andexperimental results are, respectively, denoted by openand filled data points. All of the results agree that forlow levels of noise, using this harmonic excitationamplitude, the nonlinear harvester exhibits the lowamplitude intrawell response as the underlying steadystate. Similar to the case of the role of the harmonicexcitation frequency upon the harvester electromecha-nical dynamics, the analysis used to generate the resultsof Figure 6 also suggests that coexisting dynamic beha-viors may exist for a given value of s=a. Indeed, theanalysis predicts that another low amplitude dynamicmay occur for low levels of additive noise such ass=a\1, although neither simulations nor experimentsdetect its existence. This suggests that such additionaldynamic behavior may have low likelihood of beinginduced in practice according to its basin of attraction(Harne and Wang, 2017). Nevertheless, the intrawellresponses provide negligible DC power, as shown inFigure 6(b) for low noise levels.

As the noise standard deviation is increased respect-ing the harmonic amplitude of the excitation, the DCpower gradually increases, with a significant growthobserved analytically, numerically, and experimentallyaround a ratio of s=a= 2. An explanation for thisbehavior is given by simulated and measured results inFigure 6(c) which plot the percentage of time that thebeam undergoes the coexistent snap-through dynamic.Around s=a= 2, this percentage clearly begins to

524 Journal of Intelligent Material Systems and Structures 29(4)

Figure 5. Portions of time series of experimental measurements for the (a) base accelerations provided to the energy harvesterfor harmonic-only and combined harmonic with stochastic excitations, (b, d) beam tip displacement without noise (dashed curves)and with noise s= 11:25 m=s2 (solid curves), and (c, e) voltage from piezoelectric beam and rectified voltage across resistive loadR = 47 kO, without noise (dashed curves and dotted curves) and with noise s= 11:25 m=s2 (solid curves and dash-dot curves).Results shown in (b, c) without noise correspond to the intrawell steady state indicated by the square data point in Figure 4(c), whileresults shown in (d, e) without noise correspond to the snap-through steady state indicated by the circle data point in Figure 4(c).

Figure 6. (a) Harvester harmonic displacement amplitude, (b) average charging power across resistive load R = 470 kO, and (c)percentage of time that the beam spends snapping through, as a function of the ratio between noise standard deviation and harmonicamplitude of the base excitation. Blue solid (red dash) curves indicate analyzed responses associating with snap-through (intrawell)dynamics, blue triangles are the simulated results, and green filled circles indicate experimental measurements. In all cases, theharmonic excitation amplitude is a= 2:85 m=s2 while the frequency is 13 Hz.

Dai and Harne 525

increase from near-zero values obtained for smallerratios of excitation properties. This indicates that thenoise begins to contribute favorably to DC powerdelivery provided by the harvester because the snap-through response is induced, even though the underly-ing steady-state response is a low amplitude intrawellbehavior. The results of Figure 6 suggest that for har-monic amplitudes of base excitation that are insuffi-cient to activate snap-through dynamics in the bistablenonlinear energy harvester, the DC power delivery maybe considerably enhanced by the introduced of enoughnoise, here around s=a.2, that triggers the desirabledynamic in a stochastic way, similar to the time series,as shown in Figure 5(c).

Figure 7 presents results considering greater ampli-tude of the harmonic base excitation, a= 5:85m=s2,while all other notations and system parameters remainthe same as the case shown in Figure 6. For this har-monic amplitude of base acceleration, in the absence ofnoise, the harvester exhibits the persistent snap-throughresponse as the underlying steady-state behavior, simi-lar to the time series show in Figure 5(d) and (e)derived from the measurements used to present Figure4(c) and (d). It is plainly revealed in Figure 7(a) thatthe snap-through dynamic is inhibited by low levels ofnoise, specifically when the ratio of the noise standarddeviation to amplitude of harmonic components isabout 0\s=a� 0:8. Figure 7(b) shows that the DCpower reduces greatly under such circumstances, whencompared to the harmonic-only DC power outputshown under the condition s=a= 0.

These features are illuminated further by observingthe time series of electromechanical response labeled asH and J in Figure 7(a). In Figure 8, the responselabeled as H is shown in (a) for displacement and (b)for piezoelectric and charging voltages, which considera case when the noise is insignificant in standard devia-tion when compared to the amplitude of the harmonicexcitation. When the ratio s=a is increased closer to

around 0.75, Figure 8(c) and (d) shows the correspond-ing displacement and voltage time series. Comparingthese two cases of noise contribution to the overall baseexcitation, it is evident that the snap-through dynamicis suppressed for the case of s=a’0:75, leading toreduced charging voltage and hence less DC powerdelivery due to the reduced periodicity of the mechani-cal oscillations.

Studying the results of Figure 7(a) further, when theratio s=a increases beyond around 0.75, the snap-through behavior is predicted by analysis to be the onlydynamic regime triggered. Comparatively, as shown inFigure 7(c), the percentage of time that simulations andexperiments observe the snap-through responseincreases significantly for ratios s=a greater than about0.75, which supports the more idealized prediction bythe analysis. Thus, large DC power delivery is recov-ered for greater noise contribution in the excitation.Figure 8(e) and (f) plots time series of displacementand voltage for the data point labeled as K in Figure7(a). Interestingly, the displacement is clearly not asteady snap-through, although as shown in Figure 8(f)the rectified voltage is greater than the voltage achievedwhen the persistent snap-through dynamic is triggeredin the absence of noise (Figure 8(b)). These results indi-cate that for the greater amplitude of harmonic baseexcitation, a= 5:85m=s2, that induces the snap-through response in the absence of noise, the introduc-tion of stochastic contributions to the excitation isadverse to DC power generation if the noise level is lowbecause snap-through dynamics are inhibited, but addi-tive noise may be a significant benefit to DC powerdelivery if the noise contribution is large enough, herearound s=a � 2. The theoretical predictions are ingood agreement with the trend of DC power generationfor increased noise when compared to the qualitativeand quantitative results of simulation and experiment,as shown in Figure 7(b). This suggests that the newanalytical model formulation and solution strategy

Figure 7. (a) Harvester harmonic displacement amplitude, (b) average charging power across resistive load R = 470 kO, and (c)percentage of time that the beam spends snapping through, as a function of the ratio between noise standard deviation and harmonicamplitude of the base excitation. Blue solid (red dash) curves indicate analyzed responses associating with snap-through (intrawell)dynamics, blue triangles are the simulated results, and green filled circles indicate experimental measurements. In all cases, theharmonic excitation amplitude is a= 5:85 m=s2 while the frequency is 13 Hz.

526 Journal of Intelligent Material Systems and Structures 29(4)

may be used to assist implementation of nonlinearenergy harvesters in practical application environmentswhere the base acceleration available is neither purelyharmonic nor purely stochastic.

Conclusion

In order to take advantage of the broadband and largepower generation characteristics of nonlinear energyharvesters for self-sufficient sensors in structural healthmonitoring applications, this research investigated DCpower generation from nonlinear vibration energy har-vesters subjected to excitations that contain realisticcombinations of harmonic and stochastic components.These efforts of this work established a new analyticalmethod to predict the overall electromechanicalresponses induced under such excitations, including theDC power delivery from rectifier circuits coupled tosuch nonlinear harvester platforms. With numericalverification and experimentally validation, the accuracyof the analysis is exemplified. Through subsequentmodel and experimental studies, it is found that despitethe introduction of stochastic excitation, the DC powermay be increased from steady-state levels associatedwith the low amplitude intrawell dynamic regime. Thenoise excitation contribution has more intricate influ-ences upon the snap-through responses: the additivenoise reduces the mean DC power generation at lowharmonic excitation frequencies when snap-through isachieved under pure harmonic excitations, while thestochastic excitations are beneficial when the ratio of

noise standard deviation to harmonic amplitude isgreater than about two. The outcomes indicate that theanalytical model formulation and solution strategyhave potential to assist in the design and implementa-tion of nonlinear energy harvesters in real world appli-cation for improved, practical performance. Inaddition, due to the dependence of DC power genera-tion on the resistive load, harmonic excitation charac-teristics, and additive stochastic excitation levels,optimization studies are promising directions for fur-ther investigations toward capitalizing on nonlinearenergy harvesters deployed in real world environments.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with

respect to the research, authorship, and/or publication of thisarticle.

Funding

The author(s) disclosed receipt of the following financial sup-port for the research, authorship, and/or publication of thisarticle: This research is supported by The Ohio StateUniversity Center for Automotive Research. The authors alsoacknowledge support from Mide Technology Corp.

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