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An equivalent linearization technique for nonlinearpiezoelectric energy harvesters under Gaussian white noise

Wen-An Jiang a, Li-Qun Chen a,b,c,⇑a Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, Chinab Department of Mechanics, Shanghai University, Shanghai 200444, Chinac Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China

a r t i c l e i n f o

Article history:Received 4 April 2013Received in revised form 1 December 2013Accepted 31 December 2013Available online 8 January 2014

Keywords:Piezoelectric energy harvestingEquivalent linearizationGaussian white noise

a b s t r a c t

An equivalent linearization technique is proposed to determine approximately the outputvoltage a nonlinear piezoelectric energy harvester excited by Gaussian white noise excita-tions. Equivalent linear system is derived from minimizing the mean-squared of the error.The linear equivalent coefficients are presented by the method of normal truncation. Theexact solution of equivalent linear system is derived obtained. The effectiveness of themethod is demonstrated by numerical simulations.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Energy harvesting from ambient waste energy for the purpose of running low-powered electronics has emerged as aprominent research area and continues to grow at the rapid pace. As a promising approach, the piezoelectricity has beenused to convert ambient vibration into useful electrical energy. There are several excellent and comprehensive survey papersand monographs, notably Sodano [1], Anton and Sodano [2], Priya [3], Tang et al. [4], Priya and Inman [5] and Erturk andInman [6], reviewing the state of the art in different time phases of investigations related to piezoelectric energy harvesting.

Concentrating on the resonance under a harmonic excitation, most works on energy harvesting took the deterministicapproach. Especially, nonlinearity was introduced to increase the operating frequency range of energy harvesters. Stantonet al. [7] applied the method of harmonic balance to characterize quantitatively the beam and electrical network oscillationamplitudes, and validated the response amplitudes of numerical simulations’ predicated by experiments. They investigatedthe response of harvesting energy as a nonlinear oscillator, and demonstrated that the bistability may be used to improveenergy harvesting [8]. Erturk et al. [9], Erturk and Inman [10] constructed a piezomagnetoelastic energy harvester, investi-gated numerically and experimentally the response to harmonic excitations. Triplett and Quinn [11] considered a nonlinearpiezoelectric relationship on the performance of a vibration-based energy harvester, used Poincare–Lindstedt perturbationmethod to analysis the response of the harvesting system.

Despite the successfulness of the deterministic approach, randomness, inherent in most real-world circumstances, maysignificantly change the behavior of vibration-based energy harvesters. There are some researches via stochastic approaches.Establishing the closed-form expressions for output power, proof mass, displacement, and optimal load for linear energy har-vesters driven by broadband random vibrations, Halvorsen [12] demonstrated that the output power has a different opti-mum for broadband excitations from that for sinusoidal excitations. McInnes et al. [13] employed stochastic resonance to

1007-5704/$ - see front matter � 2014 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cnsns.2013.12.037

⇑ Corresponding author at: Department of Mechanics, Shanghai University, Shanghai 200444, China. Tel.: +86 02166136905; fax: +86 2156333085.E-mail address: [email protected] (L.-Q. Chen).

Commun Nonlinear Sci Numer Simulat 19 (2014) 2897–2904

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Commun Nonlinear Sci Numer Simulat

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enhance vibration energy harvesting and revealed numerically the significant enhancement without any periodic forcing.Cottone et al. [14] found numerically and experimentally that the nonlinear oscillators can outperform the linear ones understochastic excitation. Based on a single-degree-of-freedom model, Adhikari et al. [15] analyzed the mean power of a linearpiezoelectric energy harvester under stationary Gaussian white noise. Gammaitoni et al. [16] revealed nonlinear oscillatorscan outperform the linear ones under noise excitation in monostable configurations. However, Daqaq [17] demonstratedmonostable Duffing oscillator does not provide any enhancement over the typical linear oscillators under white Gaussianand colored excitations. Litak et al. [18] calculated the response of a nonlinear piezomagnetoelastic energy harvester understationary Gaussian white noise. Daqaq [19] derived an approximate expression for the mean power under exponentiallycorrelated noise and demonstrated the existence of an optimal potential shape maximizing the output power. Analyzinga piezomagnetoelastic energy harvester of strongly nonlinear under random excitations, Ali et al. [20] established aclosed-form approximate expression of the harvested power and validated against numerical Monte Carlo simulation results.Zhu and Zu [21] presented a new magnetoelectric energy harvester from nonlinear vibrations by magnetic levitation, derivedthe government equations of the model and obtained the mechanical and electrical responses in time-domain. Green et al.[22] reported Duffing-type nonlinearities can reduce the size of energy harvesting devices without affecting their power out-put, verified the result using the technique of equivalent linearization.

Among various approaches nonlinear random vibration, the equivalent linearization is a simplest in principle and easiestin practice. The basic idea of equivalent linearization techniques is to replace a given nonlinear stochastic system, for whichthe exact stationary solution is not obtainable analytically, with a linear stochastic system, whose behavior is closest to thegiven one in some statistical sense and whose exact stationary solution is obtainable. It is becoming a convenient approx-imate approach to predict the stationary response of nonlinear stochastic systems. Historically, Caughey [23] proposedthe equivalent linearization technique for a nonlinear oscillator subjected to stationary Gaussian random excitation. Iwanand Yang [24] extended the equivalent linearization technique to nonlinear multi-degree-of-freedom systems. Spanos[25] developed stochastic linearization for symmetric or asymmetric nonlinear systems, outlined the solution procedurefor determining nonstationary or stationary system response statistics.

It should be remarked that there are different types of electrical circuit equations used in piezoelectric energy harvesting.duToil et al. [26] first proposed a coupled electromechanical equation for lumped-parameter piezoelectric energy harvesters.As the motion equation of the harvester and its electromechanical equation cannot be directly decoupled, the system is with1.5 degrees-of-freedom. This type of the electrical circuit equation has been widely used [7–11,12–16,18,20]. Daqaq [17,19]and Green et al. [22] introduced an uncoupled electrical circuit equation, the electromechanical equation can be convertedinto first-order differential equation. Therefore, the system is with single-degree-of-freedom. Triplett and Quinn [11] consid-ered a nonlinear piezoelectric coupling relationship on the performance of a vibration-based energy harvester. Adhikari et al.[15] reported an electrical circuit equation with an inductor, where the electrical equation is second-order differential equa-tion. Thus, the system is with 2 degrees-of-freedom. This paper investigated the lumped-parameter model of a piezoelectricenergy harvester which is essentially a 1.5 degree-of-freedom system. So far, to the authors’ best knowledge, there is noequivalent linearization analysis on such a system. To address the lacks of research in the aspect, the present work developsthe equivalent linearization technique to determine the response of nonlinear piezoelectric energy harvesters under Gauss-ian white noise excitation.

The paper is organized as follows. Section 2 presents the governing equation of piezoelectric coupling systems underGaussian white noise excitation. Section 3 derives the equivalent linear system from the criteria of least mean-squared error,and gives the exact solutions of equivalent linear system. Section 4 analyzes the validity of the method via numerical sim-ulations. Section 5 ends the paper with concluding remarks.

2. The governing equation

The lumped-parameter model of a piezoelectric energy harvester with cubic nonlinearity in the displacement term underGaussian white noise excitation can be given as

€xþ 2f _xþx20xþ eax3 � vt ¼ nðtÞ; ð1Þ

_tþ ktþ j _x ¼ 0; ð2Þ

where x is the displacement response, t is the voltage response across the external electrical load, x0 is the undamped fun-damental natural frequency, v is the piezoelectric coupling term in the mechanical equation, k is the reciprocal of the timeconstant of the resistive–capacitive circuit, j is the piezoelectric coupling term in the electrical equation. Furthermore, e is asmall bookkeeping parameter and f is a mechanical damping term, nðtÞ is stationary Gaussian white noise process with zeromean and autocorrelation function

hn tð Þn tð þ sÞi ¼ 2pS0dðsÞ; ð3Þ

where hi denotes the expected value, S0 is the spectral density of the excitation, and d is the Dirac function.

2898 W.-A. Jiang, L.-Q. Chen / Commun Nonlinear Sci Numer Simulat 19 (2014) 2897–2904


3. Equivalent linearization technique

It will be assumed that a stationary solution of Eqs. (1) and (2) exist. Define the following set of linear equations

€xþ ð2fþ ceÞ _xþ x20 þ ke

� �x� vt ¼ nðtÞ; ð4Þ

_tþ ktþ j _x ¼ 0; ð5Þ

where ce and ke are damping and stiffness coefficients to be determined by equivalent linearization to replace the nonlinearterm eax3. The replacement of a nonlinear system by a linear system will yield the error. The error may be defined as thedifference of the two systems. Considering Eqs. (1) and (4), one can write the error as

e ¼ eax3 � ce _x� kex; ð6Þ

To approximate the solution of the nonlinear system (1) and (2), ce and ke should be selected such that error e would be assmall as possible. Since in Eq. (6) x is a random variable, e will also be a random variable, and thus its smallness is subject tothe different interpretations. A criterion used here is to require that the mean square value of e is a minimum, i.e.,

@E½e2�@ce

¼ @E½e2�@ke

¼ 0: ð7Þ

Substituting Eq. (6) into Eq. (7) and performing the partial differentiations in the resulting equation, yield

eaE½ _xx3� � ceE½ _x2� � keE½ _xx� ¼ 0; ð8Þ

eaE½x4� � ceE½ _xx� � keE½x2� ¼ 0; ð9Þ

Solving Eqs. (8) and (9) yields the equivalent linear coefficients

ce ¼eaE½ _xx3�E½x2� � eaE½x4�E½x _x�

E½x2�E½ _x2� � ðE½x _x�Þ2; ke ¼

eaE½x4�E½ _x2� � eaE½ _xx3�E½x _x�E½x2�E½ _x2� � ðE½x _x�Þ2

: ð10Þ

Equivalent linear coefficients were determined under stationary response, As the stationary displacement and the sta-tionary velocity are uncorrelated, i.e., E½x _x� ¼ 0, therefore

ce ¼eaE½ _xx3�

E½ _x2� ; ke ¼eaE½x4�

E½x2� : ð11Þ

Since the function x3 is satisfied the theorem of normal truncation [27], the higher term of the right hand side of Eq. (11)can be expressed in terms of second order moments by applying repeatedly the method of normal truncation [27],

ce ¼ eaE@x3

@ _x

� �¼ 0; ke ¼ eaE

@x3

@x

� �¼ 3eaE½x2�: ð12Þ

Transformation of linear Eqs. (4) and (5) into the frequency domain gives

�x2 þ ix2fþ ðx20 þ keÞ

� �XðxÞ � vVðxÞ ¼ NðxÞ; ð13Þ

jixXðxÞ þ ðixþ kÞVðxÞ ¼ 0: ð14Þ

where XðxÞ, VðxÞ, NðxÞ are, respectively, the Fourier transforms of xðtÞ, tðtÞ and nðtÞ. From Eqs. (13) and (14), the displace-ment and the voltage in the frequency domain are

XðxÞ ¼ ixþ kW

NðxÞ; ð15Þ

VðxÞ ¼ �jixW

NðxÞ; ð16Þ

where WðixÞ ¼ ðixÞ3 þ ð2fþ kÞðixÞ2 þ ð2kfþx20 þ ke þ jvÞixþ kðx2

0 þ keÞ .Then the second order moment given as

E½x2� ¼Z 1

�1UxxðxÞdx ¼

Z 1

�1

ixþ kW

� �2

Unndx ¼ S0

Z 1

�1

k2 þx2

WðxÞW�ðxÞdx: ð17Þ

The calculation of the integral on the right-hand side of Eq. (17) can be cast into a general form

In ¼Z 1

�1

DnðxÞKnðxÞK�nðxÞ

dx ð18Þ

W.-A. Jiang, L.-Q. Chen / Commun Nonlinear Sci Numer Simulat 19 (2014) 2897–2904 2899


where the polynomials are

DnðxÞ ¼ bn�1x2ðn�1Þ þ bn�2x2ðn�2Þ þ � � � þ b0;

KnðxÞ ¼ anðixÞn þ an�1ðixÞn�1 þ � � � þ a0:

Following Roberts and Spanos [28], one evaluates this integral as

In ¼pan

det½Bn�det½An�

ð19Þ

where the n� n matrices are defined as

Bn ¼

bn�1 bn�2 0 � � � 0 b0

�an an�2 �an�4 � � � 0 00 �an�1 an�3 � � � 0 00 an �an�2 � � � 0 0� � � � � � � � � � � � � � � � � �0 0 0 � � � �a2 a0

2666666664

3777777775

and An ¼

an�1 �an�3 an�5 � � � 0 0�an an�2 �an�4 � � � 0 0

0 �an�1 an�3 � � � 0 00 an �an�2 � � � 0 0� � � � � � � � � � � � � � � � � �0 0 0 � � � �a2 a0

2666666664

3777777775:

Now based on Eq. (19), the integral (17) can be evaluated as

I3 ¼det½B3�det½A3�

p ¼ ð2fþ kÞk2 þ kðx20 þ keÞ

kðx20 þ keÞ½ð2fþ kÞð2kfþx2

0 þ ke þ jvÞ � kðx20 þ keÞ�

p

¼ 2fkþ k2 þx20 þ ke

ðx20 þ keÞ½2fke þ ð2fþ kÞð2kfþx2

0 þ jvÞ � kx20�

p; ð20Þ

where

B3 ¼0 1 k2

�1 2kfþx20 þ ke þ jv 0

0 �2f� k kðx20 þ keÞ

264

375

and A3 ¼2fþ k �kðx2

0 þ keÞ 0�1 2kfþx2

0 þ ke þ jv 00 �2f� k kðx2

0 þ keÞ

264

375:

From Eqs. (12), (17), and (20), one has

ak3e þ bk2

e þ cke ¼ d; ð21Þ

where

a ¼ 2f; b ¼ 2fx20 þ 2fþ kð Þð2kfþx2

0 þ jvÞ � kx20;

c ¼ x20 ð2fþ kÞð2kfþx2

0 þ jvÞ � kx20

� �� 3eapS0; d ¼ 3eapS0 2fkþ k2 þx2

0

� �:

Solving Eq. (21), one can obtains the equivalent linear coefficient ke.Linear Eqs. (4) and (5) can be converted to the following set of first-order equations

_zðtÞ ¼ AzðtÞ þ f ðtÞ; ð22Þ

where

zðtÞ ¼ ½ xðtÞ _xðtÞ tðtÞ �T ; f ðtÞ ¼ ½0 nðtÞ 0 �T ; and A ¼0 1 0

�ðx20 þ keÞ �2f v0 �j �k

264

375:

Characteristic equation of homogeneous term of system (22) is

detðsE � AÞ ¼sE �1 0

x20 þ ke sEþ 2f �v

0 j sEþ k

264

375 ¼ s3 þ c1s2 þ c2sþ c3 ¼ 0; ð23Þ

One can get eigenvalues s1; s2; s3 and corresponding eigenvectors u1;u2;u3, respectively. Here c1; c2 and c3 are constants,and ui ¼ ½u1i u2i u3i �T ði ¼ 1;2;3Þ. Then the basic solution matrix of homogeneous equation is



eAt ¼ /ðtÞ/�1ð0Þ; ð24Þ

where /ðtÞ ¼ ½ es1tu1 es2tu2 es3tu3 �. It is easy to get the solutions of non-homogeneous Eq. (22)

zðtÞ ¼ eAðt�t0Þzð0Þ þZ t

t0

eAðt�sÞf ðsÞds: ð25Þ

where zð0Þ are initial values.

4. Numerical simulations

The response a nonlinear piezoelectric energy harvester can be investigated numerically by simulation of Eqs. (1) and (2)via Milstein scheme. The system parameters are chosen as follows, f ¼ 0:01;x0 ¼ 1;a ¼ 5;v ¼ 0:05; k ¼ 0:05;j ¼ 0:5;2pS0 ¼ 0:01. The initial conditions are set at the equilibrium state, namely, xð0Þ ¼ 0; _xð0Þ ¼ 0; tð0Þ ¼ 0. Fore = 0.01, 0.02, 0.05 and 0.1, Figs. 1–4 illustrate the displacement response and the output voltage response of original non-linear system (1) and (2), equivalent linearization system (4) and (5), and the direct linearization by omitting the nonlinearterm (inserting a = 0 in Eqs. (1) and (2)). In Figs. 5–8 the response differences between the original system and the equivalentlinearization are compared with those between the original nonlinear system and the direct linearization. The numerical re-sults show that the errors are rather small for e = 0.01 and 0.02 while quite large for e = 0.1, the errors increases with thenonlinearity, and the equivalent linearization errors are smaller than the direct linearization errors.

Fig. 1. Displacement and voltage responses for e ¼ 0:01.






Fig. 5. Displacement and voltage errors for e ¼ 0:01.




Fig. 7. Displacement and voltage error for e ¼ 0:05.




5. Conclusions

This paper is devoted to responses of nonlinear piezoelectric energy harvesters to Gaussian white noise excitation. Anequivalent linearization technique is proposed to determine approximately the responses. The equivalent linear system isderived from the criteria of least mean-squared error. The linear equivalent coefficients are obtained by the method of nor-mal truncation. The exact solution of equivalent linear system is obtained. The displacement and the output voltage re-sponses are numerically calculated for the original nonlinear system, the equivalent linearization, and the directlinearization. The simulations demonstrate that the equivalent linearization approximate the original nonlinear system wellfor the weak nonlinearity, and better than the direct linearization.

Acknowledgments

This work was supported by the State Key Program of National Natural Science of China (No. 11232009) and ShanghaiLeading Academic Discipline Project (No. S30106).

References

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