Contents lists available at ScienceDirect

Journal of Sound and Vibration

Journal of Sound and Vibration 377 (2016) 264–283

http://d0022-46

n Corrfax: þ8

E-m

journal homepage: www.elsevier.com/locate/jsvi

Stochastic averaging based on generalized harmonic functionsfor energy harvesting systems

Wen-An Jiang a, Li-Qun Chen a,b,c,n

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 8 December 2015Received in revised form15 April 2016Accepted 4 May 2016

Handling Editor: L. G. Tham

function is given through the equivalent potential energy which is independent of the

Keywords:Vibratory energy harvestingnonlinearityStochastic averagingGeneralized harmonic transformationMonte Carlo simulation

x.doi.org/10.1016/j.jsv.2016.05.0120X/& 2016 Elsevier Ltd. All rights reserved.

esponding author at: Department of Mech6 21 66134463.ail address: lqchen@staff.shu.edu.cn (L.-Q. Ch

a b s t r a c t

A stochastic averaging method is proposed for nonlinear vibration energy harvesterssubject to Gaussian white noise excitation. The generalized harmonic transformationscheme is applied to decouple the electromechanical equations, and then obtained anequivalent nonlinear system which is uncoupled to an electric circuit. The frequency

total energy. The stochastic averaging method is developed by using the generalizedharmonic functions. The averaged Itô equations are derived via the proposed procedure,and the Fokker-Planck-Kolmogorov (FPK) equations of the decoupled system are estab-lished. The exact stationary solution of the averaged FPK equation is used to determine theprobability densities of the amplitude and the power of the stationary response. Theprocedure is applied to three different type Duffing vibration energy harvesters underGaussian white excitations. The effects of the system parameters on the mean-squarevoltage and the output power are examined. It is demonstrated that quadratic non-linearity only and quadratic combined with properly cubic nonlinearities can increase themean-square voltage and the output power, respectively. The approximate analyticaloutcomes are qualitatively and quantitatively supported by the Monte Carlo simulations.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Energy harvesting from mechanical energy to support low-consumed electronics has arose as an outstanding researchfield and continues to grow rapidly. There are several prominent and comprehensive review papers and monographs,especially Tang et al. [1], Pellegrini et al. [2], Harne and Wang [3], Daqaq et al. [4], Erturk and Inman [5] and Elvin and Erturk[6], introducing the development and the situation of energy harvesting.

To enhance the broadband feature of vibration-based energy harvesters (VEHs), researchers have proposed severalprominent strategies, such as linear resonance frequency tuning [7], multimodal VEHs [8], and stiffness nonlinearitiescharacteristics [3,4]. Mann and Sims [9] designed a novel energy harvesting mechanism which uses a magnet to generate anonlinear restoring force, employed the method of multiple scales to determine the response, and validated the response viathe experimental data. Erturk and Inman [10] constructed a piezomagnetoelastic energy harvester and obtained numerically

anics, Shanghai University, 99 Shang Da Road, Shanghai 200444, China. Tel.: þ86 21 66136905;

en).

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283 265

and experimentally the voltage response under harmonic excitation. Zhu and Zu [11] designed a buckled-beam piezoelectricenergy harvester to improve the bandwidth of the output voltage by introducing a midpoint magnetic force. Zhou et al. [12]presented numerical and experimental data of a bistable piezomagnetoelastic energy harvester with rotatable magnets thatimproved broadband frequency voltage response.

Since randomness inherent in most real-world environments may essentially change the characteristic of energy har-vesters, many researchers treated vibration-based energy harvesting under random excitations with all sorts of stochasticapproaches. Cottone et al. [13] found the bistable energy harvester is superior to the linear ones under stochastic excitationvia numerical simulation and experiment. However, Daqaq [14] illustrated that monostable Duffing oscillator cannot affordany enhanced power than the relevant linear system under Gaussian white noise and colored noise excitations. Daqaq [15]derived an approximate expression for the mean power of vibration energy harvester subject to exponentially correlatednoise and proved that there is an optimal potential shape leading to maximize the output power. Green et al. [16] reportedthat Duffing-type nonlinearities of electromagnetic energy harvesting can reduce their output power and verified thetendency using the technique of equivalent linearization. Ali et al. [17] established a closed-form approximate powerexpression of bistable piezoelectric energy harvester under random excitation, and validated the analytical predict valuesagainst the Monte Carlo simulation results. Masana and Daqaq [18] calculated the voltage response of buckling piezoelectricbeam under band-limited noise, discussed the influence of stiffness-type nonlinearities on the displacement and voltageresponse. Daqaq [19] presented the voltage response statistics by introducing the method of moment differential equationsof FPK equation and demonstrated that the time constant ratio of the energy harvester plays a key role in developing theperformance of nonlinear harvesters under the random environment. He and Daqaq [20,21] employed the statistical line-arization techniques and finite element method of the FPK equation to investigate the effects of the potential energyfunction on the mean steady-state approximate output power. Xu et al. [22] proposed a novel decoupling technique todevelop a stochastic averaging of energy envelope for Duffing-type vibration-based energy harvesters, and discussed theeffects of the system parameters on the mean square output voltage and power. Kumar et al. [23] used the finite elementmethod to solve the FPK equation of the associated bistable energy harvester, and analyzed the effects of the systemparameters on the mean square output voltage and power. Jin et al. [24] introduced the generalized harmonic transfor-mation to decouple the electromechanical equations, and applied the equivalent nonlinearization technique to derive asemi-analytical solution of the corresponding nonlinear vibration energy harvesters subjected to Gaussian white noiseexcitation.

The Fokker–Planck–Kolmogorov equation provides a powerful tool for treating the statistical characteristics of nonlinearstochastic system. The solutions of Fokker–Planck–Kolmogorov equation define the time-dependent evolutions of prob-ability densities. However, exact solutions are usually difficult to obtain for nonlinear stochastic systems. In fact, they havebeen found only in a few special cases. The stationary solution, namely, the steady-state probability density, is relativelyobtainable and useful in some practical circumstances. The FPK equation of the coupled electromechanical system is a three-dimensional nonlinear partial differential equation. Its exact solution is difficult to obtain, even for exact stationary prob-ability densities. To date all known exact stationary solutions of the FPK equation have been obtained only for degradedcases of the decoupled electromechanical system [14–16]. Therefore some approximate methods for solving the FPKequation of the coupled electromechanical system have been reported which included the statistical linearization techni-ques [17,20,21], the moment differential equations method [19], the finite element method [20,21,23], the stochasticaveraging of energy envelope [22], the equivalent nonlinearization technique [24].

Among various approaches to nonlinear random vibration, the stochastic averaging method is a powerful approximatetechnique for the prediction of response of nonlinear system subject to external and parametric random excitations. Thesuccess of the stochastic averaging method is mainly due to the reduction of dimensions of the FPK equation while theessential behavior of the system is retained. It is also a convenient approximate approach to predict the stationary responseof nonlinear stochastic systems and has been extensively used in theory and engineering application of random vibration.There are several excellent and comprehensive review papers, observably Roberts [25], Crandall and Zhu [26] and Zhu [27],reviewing the stochastic averaging methods in different times. Roberts [28] employed the stochastic averaging method toinvestigate the response of ship rolling motion and obtained the exact stationary probability density function. Roberts andSpanos [29] applied the stochastic averaging method to study the response of nonlinear oscillator under external excitationwith or without combined parametric excitation, and obtained the exact stationary and non-stationary probability densityfunction. Huang et al. used the generalized harmonic functions averaging method to predict the response of Duffing-van derPol oscillator under combined harmonic and white-noise excitations [30] and wide-band random excitation [31]. So far, tothe authors’ best knowledge, there is no the generalized harmonic functions stochastic averaging analysis on energy har-vesting. To address the lacks of research in this aspect, the present work develops the stochastic averaging technique todetermine the response of nonlinear energy harvesters under Gaussian white noise excitation. The developed approach canquantitatively account for the effects of nonlinearities on the mean-square voltage and the output power. It may alsoprovide a foundation for further qualitative investigations such as bifurcation analysis, while bifurcations are not treated.

The paper is organized as follows. Section 2 reviews the generalized harmonic functions of a nonlinear conservativeoscillator. Section 3 introduces the basic model of nonlinear vibration energy harvesters. Section 4 decouples the electriccircuit system and the mechanical system of nonlinear vibration energy harvesters. Section 5 presents the stochasticaveraging procedure of nonlinear vibration energy harvesters by using the generalized harmonic functions. Sections 6–8

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283266

apply the stochastic averaging method to three different type Duffing vibration energy harvesters. Section 9 ends the paperwith concluding remarks

2. Generalized harmonic functions

Consider the free vibration of a nonlinear conservative oscillator. The equation of motion of the system can be written as

€xþgðxÞ ¼ 0 (1)

The first energy integral of the system is

12_x2þVðxÞ ¼H (2)

where H is the total energy and the potential energy is

VðxÞ ¼Z x

0gðxÞdx (3)

Assume that Eq. (1) has periodic solutions surrounding the equilibrium point in domain. The periodic solution of Eq. (1)is of the form [32]

xðtÞ ¼ a cos φðtÞþb; _xðtÞ ¼ �aωða;φÞ sin φðtÞ (4)

where φðtÞ ¼ ψðtÞþθ, and

ωða;φÞ ¼ dψdt

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 VðaþbÞ�Vða cos φþbÞ� �

a2 sin 2φ

s(5)

where a and b are constants, and

VðaþbÞ ¼ Vð�aþbÞ ¼H (6)

and cosφ and sinφ are called the generalized harmonic functions [32], and a is the amplitude of displacement and ω (a, φ) isthe frequency of the system.

Expanding ω�1 (a, φ) into Fourier series

ω�1ða;φÞ ¼ C0aφþX1n ¼ 1

1nCnðaÞ sin nφ (7)

Substituting Eq. (7) into Eq. (5), then integrating Eq. (5) yields

t ¼ C0aφþX1n ¼ 1

1nCnðaÞ sin nφ (8)

Further integrating Eq. (8) with respect to φ from 0 to 2π leads to the averaged period and the averaged frequency

TðaÞ ¼ 2πC0ðaÞ; ωðaÞ ¼ 1=C0ðaÞ (9)

3. Basic model of VEH

Fig. 1 shows a simplified model of single degree of freedom nonlinear VEH which contains a mechanical oscillatorcoupled to an electric circuit equation via an electromechanical coupling conversion mechanism. For example, piezoelectric

Fig. 1. A simplified model of the vibratory energy harvester [20]. (a) Piezoelectric and (b) Electromagnetic.

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283 267

energy harvester, as shown in Fig. 1(a), or electromagnetic energy harvester, as indicated in Fig. 1(b). The electromechanicalcoupling equations of motion can be expressed as

m€xþc_xþdUðxÞdx

þχy¼ �m€xb (10a)

Cp _yþy=R¼ χ _xðpiezoelectricÞ; L_yþRy¼ χ _xðelectromagneticÞ (10b)

where x represents the displacement of the mass, c is the damping coefficient, χ is the linear electromechanical couplingcoefficient, €xb is the base acceleration, Cp is the capacitance, L is the inductance, and y is the electric quantity representingthe induced voltage in piezoelectric harvesters and the induced current in the electromagnetic ones. These are measuredacross an equivalent resistive load R.

The non-dimensional electromechanical coupling equations can be obtain [20,21]

€Xþ2ζ _XþgðXÞþκY ¼ ξðtÞ (11a)

_YþαY ¼ _X (11b)

where gðxÞ ¼ dUðxÞ=dx, and ξðtÞ ¼ � €xb is the base acceleration. ξðtÞ is Gaussian white noise with zero mean and auto-correlation function

ξðtÞξðtþτÞ� �¼ 2DδðτÞ (12)

in which hi denotes the expected value, D is the intensity of the excitation, and δðτÞ is the Dirac function.

4. Decoupling of the electromechanical coupling system

Using the generalized harmonic function [32], suppose the mechanical states of Eq. (11) can be written as

XðtÞ ¼ AðtÞ cos ΦðtÞþFðtÞ (13a)

_XðtÞ ¼ �AðtÞωðA;ΦÞ sin ΦðtÞ (13b)

ΦðtÞ ¼ ψðtÞþΘðtÞ (13c)

where

ωðA;ΦÞ ¼ dψ=dt (14)

and F(t) is random process and related to the total energy H [31]. Substitution of Eq. (13) into Eq. (11b), and integratingyields

YðtÞ ¼ CðtÞe�αtþ AωðA;ΦÞα2þω2ðA;ΦÞ ωðA;ΦÞ cos ϕ�α sin ϕð Þ

¼ CðtÞe�αtþ ω2ðA;ΦÞα2þω2ðA;ΦÞXðtÞþ

α

α2þω2ðA;ΦÞ_XðtÞ (15)

where C(t) is unknown function and can be determined under the initial condition. Substituting Eq. (15) into Eq. (11a), oneobtains the equivalent uncoupled mechanical equation

€Xþς _Xþ f ðXÞ ¼ ξðtÞ (16)

where

ς¼ 2ζþ ακ

α2þω2ðA;ΦÞ; f ðXÞ ¼ω2ðA;ΦÞκ

α2þω2ðA;ΦÞXðtÞþgðXÞ:

The frequency function can be obtained through the equivalent potential energy

ωðA;ΦÞ ¼ dψdt

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 V�ðAþFÞ�V�ðA cos ΦþFÞ� �

A2 sin 2Φ

s(17)

where the equivalent potential energy is

V� ¼Z X

0f ðuÞdu (18)

It is worth noting that Xu et al. [22] first proposed a decoupling method which needs the integration by parts, thevariable transformation and the triangle function expansion based on the assumption of slow-varying characteristic of themechanical energy and the initial phase. Here the approximate voltage expression is directly derived from the solution ofthe displacement. In addition, the frequency function depends on the potential energy only. For a special equivalent

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283268

potential energy, one can obtain the frequency function from Eq. (17), which seems a bit simpler than the energy-dependentfrequency function [22].

5. Stochastic averaging procedure

The equivalent nonlinear system (Eq. (16)) can be solved by using approximate technique for small damping and weekrandom excitation. The uncoupled mechanical equation can be written as

€Xþ f ðXÞ ¼ �ες _Xþ ffiffiffiε

pξðtÞ (19)

Substitution Eq. (13) into Eq. (19) yields,

dAðtÞdt

¼ εm1ðA;Φ; tÞþffiffiffiε

pσ1ðA;Φ; tÞξðtÞ

dΦðtÞdt

¼ εm2ðA;Φ; tÞþffiffiffiε

pσ2ðA;Φ; tÞξðtÞ (20)

where

m1ðA;Φ; tÞ ¼ � ςA2ω2ðA;ΦÞf ðAþFÞð1þhÞ sin

2ΦðtÞ;m2ðA;Φ; tÞ ¼ � ςAω2ðA;ΦÞf ðAþFÞð1þhÞ sin ΦðtÞ cos ΦðtÞ

σ1ðA;Φ; tÞ ¼ � AωðA;ΦÞf ðAþFÞð1þhÞ sin ΦðtÞ; σ2ðA;Φ; tÞ ¼ � ωðA;ΦÞ

f ðAþFÞð1þhÞ cos ΦðtÞ (21)

and

h¼ dFdA

¼ f ð�AþFÞþ f ðAþFÞf ð�AþFÞ� f ðAþFÞ (22)

On the basis of a theorem due to Khasminskii [33], ðA;ΦÞ are two dimensional diffusive Markov processes approximately.According to the stochastic averaging method, the averaged drift and diffusion coefficients are,

a1ðA;Φ; tÞ ¼ εm1ðA;Φ; tÞþεDσ1∂σ1∂A

þεDσ2∂σ1∂Φ

a2ðA;Φ; tÞ ¼ εm2ðA;Φ; tÞþεDσ1∂σ2∂A

þεDσ2∂σ2∂Φ

b11ðA;Φ; tÞ ¼ ε2Dσ21ðA;Φ; tÞ; b22ðA;Φ; tÞ ¼ ε2Dσ22ðA;Φ; tÞb12ðA;Φ; tÞ ¼ b21ðA;Φ; tÞ ¼ ε2Dσ1ðA;Φ; tÞσ2ðA;Φ; tÞ (23)

Appling the deterministic averaging method applied to Eq. (23) yields the following expression

a1ðAÞ ¼12π

Z 2π

0a1ðA;Φ; tÞdΦ; a2ðAÞ ¼

12π

Z 2π

0a2ðA;Φ; tÞdΦ;

b11ðAÞ ¼12π

Z 2π

0b11ðA;Φ; tÞdΦ; b12ðAÞ ¼

12π

Z 2π

0b12ðA;Φ; tÞdΦ;

b22ðAÞ ¼12π

Z 2π

0b22ðA;Φ; tÞdΦ (24)

After the stochastic averaging and the deterministic averaging, ðA;ΦÞ are homogeneous diffusive processes, approxi-mately. Therefore, Eq. (20) can be generated by expressing the equations in the Itô stochastic form as

dAðtÞ ¼ a1ðAÞdtþffiffiffiffiffiffiffiffiffiffiffiffiffib11ðAÞ

qBðtÞ (25a)

dΦðtÞ ¼ a2ðAÞdtþffiffiffiffiffiffiffiffiffiffiffiffiffib22ðAÞ

qBðtÞ (25b)

where B(t) is the unit Wiener process.Duo to Eq. (25a) is independent of ΦðtÞ, AðtÞ is homogeneous diffusive process approximately. So the FPK equation

associated with Eq. (25a) is

∂p∂t

¼ � ∂∂A

ða1pÞþ12

∂2

∂A2ðb11pÞ (26)

and the initial condition of Eq. (26) is

p¼ δðA�A0Þ; t ¼ 0 (27)

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283 269

the stationary solution of FPK Eq. (26) for system (19) is of the form

pðaÞ ¼ C

b11exp

Z a

0

2a1ðuÞb11ðuÞ

du

" #(28)

where C is a normalized constant from the following integration C ¼ b11R10 exp

R a0

2a1ðuÞb11ðuÞ

duh i

dan o�1

. The stationary prob-ability density of total energy or energy envelope can be obtained

pðHÞ ¼ pðaÞ dadH

��������¼ pðaÞ

f ðaÞ (29)

Furthermore, the stationary probability density of displacement and velocity can be obtained from p(H) as follows

pðX; _XÞ ¼ pðHÞTðHÞ

����H ¼ _X

2=2þV�ðXÞ

(30)

The non-dimensional mean-square voltage is then derived through the approximate relation in Eq. (15)

E V2h i

¼ Eω4ðAÞ

α2þω2ðAÞ� 2X2

" #þ2E

αω2ðAÞα2þω2ðAÞ� 2X _X

" #þE

α2

α2þω2ðAÞ� 2 _X2" #

(31)

Then the non-dimensional output power is expressed as

P ¼ ακE½V2� (32)

Some applications of the procedure are as follows.

6. Applied to Duffing-type VEH

The Duffing-type VEH is an important model of nonlinear vibration energy harvester, has already been widely reported[9,11,18,19,22,24]. Examples of such VEHs include, but are not limited to, the magnetically levitated inductive harvesterproposed by Mann and Sims [9] and the axially loaded piezoelectric energy harvester presented by Masana and Daqaq [18](Fig. 2).

Based on the procedure of Section 3, one obtain the following non-dimensional electromechanical coupling equations

€Xþ2ζ _XþXþδX3þκY ¼ ξðtÞ (33a)

_YþαY ¼ _X (33b)

The conservative system associated with Eq. (33a) has periodic solution of form (13). According to Eq. (6) and Section 4,we can obtain FðtÞ ¼ h¼ 0;and

f ðXÞ ¼ ω2ðA;ΦÞκα2þω2ðA;ΦÞþ1

�XþδX3;V�ðXÞ ¼ 1

2ω2ðA;ΦÞκ

α2þω2ðA;ΦÞþ1 �

X2þ14δX4 (34)

Using Eq. (17), one obtains the frequency function equation

ω4�Γ1ω2�Γ0 ¼ 0 (35)

where

Γ1 ¼ κ�α2þ1þδA2ð3þ cos 2ΦÞ=4;Γ0 ¼ 1þδA2ð3þ cos 2ΦÞ=4h i

α2:

and the corresponding frequency function

ωðA;ΦÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΓ1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΓ21þ4Γ0

q2

vuut(36)

Fig. 2. Schematics of different Duffing-type VEH [9,18].

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283270

ωðA;ΦÞcan be approximatively obtained by the following finite Fourier series

ωðA;ΦÞ ¼ω0ðAÞþω1ðAÞ cos 2Φþω2ðAÞ cos 4Φ (37)

Consequently, the averaged frequency can be approximately written as

ωðAÞ ¼ω0ðAÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiκþ1þδA2

q(38)

Substituting Eqs. (13) and (15) into Eq. (33), adopting the same method as used in Section 5, and yields

a1ðAÞ ¼Aðκþ1þδA2Þ2ðκþ1þδA2Þ

2ζþ ακ2

κþ1þδA2

� � DδA

2ðκþ1þδA2Þ2þ D

2Aðκþ1þδA2Þ;

a2ðAÞ ¼ 0; b11ðAÞ ¼Dðκþ1þδA2Þðκþ1þδA2Þ2

;

b12ðAÞ ¼ b21ðAÞ ¼ 0; b22ðAÞ ¼Dðκþ1þδA2Þ½ðκþ1ÞAþδA3�2

(39)

The stationary probability density of amplitude of Eq. (33) is in the form

pðaÞ ¼ Cðkaþδa3Þexp �ξðka2þ0:5δa4Þ=ð2DÞ� �(40)

where C is a normalization constant as defined after Eq. (28) and k¼ κ=ðα2þ1Þþ1; ξ¼ 2ζþακ=ðα2þ1Þ. The stationaryprobability density of displacement and velocity can be further obtained from p(H) as follows

pðx; _xÞ ¼ Cexp �ξð_x2þkx2þ0:5δx4Þ=ð2DÞh i

(41)

Using Eqs. (31), (32) and (41), one obtains the mean-square voltage and the power.To understand the effects of the system parameters on the expected stationary output power, variations of the stationary

mean square voltage and the output power will be investigated for different system parameters. For specified parameters,the mean square voltage and the output power can be determined by Eqs. (31) and (32), respectively. For specified para-meters and initial conditions, Eqs. (33) can be numerically integrated via the Stochastic Communication Toolbox in Matlabthrough the Euler–Maruyama algorithm [23] with a time step of 0.001 and 50 sample trajectories. The stationary probabilitydensity of the Eq. (33) can be calculated through the Monte-Carlo simulations to evaluate the accuracy of the proposedmethod. In the following, the solid lines are the analytical results of stochastic averaging method, the solid circles arenumerical results based on the governing equations. The system parameters listed in Table 1 and the initial conditions areset as the static equilibrium position Xð0Þ ¼ 0; _Xð0Þ ¼ 0;Yð0Þ ¼ 0. It should be remarked that the responses may depend oninitial conditions, while for energy harvesting problems the zero initial conditions make sense.

The probability densities of the displacement, the velocity, and the amplitude determined by Eqs. (40) and (41) arecompared with those by Monte Carlo numerical simulations of Eq. (33) in Fig. 3. Fig. 3 demonstrate that the results obtainedby using the proposed stochastic averaging method are in good agreement with those from Monte Carlo numericalsimulations. On the other hand, it is confirmed that if the input signal of dynamical system is Gaussian white noise exci-tation then the output response of the displacement and the velocity are Gaussian distribution, and the amplitude isRayleigh distribution.

Figs. 4–8 depict the variation of the mean square voltage and the output power with excitation intensity D, the dampingcoefficient ζ, the cubic nonlinear coefficient δ, the piezoelectric coupling coefficientκ, the time constant ratio α. It was clearthat the increasing excitation intensity and the decreasing damping coefficient lead to the increasing mean square voltageand increasing output power.

Fig. 6 shows that the mean square voltage and the output power decrease with the cubic nonlinear coefficient δ increase.Fig. 7 indicates that the mean square voltage decreases with the coupling coefficient increases, and there exists an optimalpiezoelectric coupling coefficient to obtain the maximal output power. Fig. 8 demonstrates that there exists an optimal timeconstant ratio to obtain the maximal output power, which means that there exists an optimal resistance to generate the

Table 1Values for parameters used in simulations.

Parameters Values

Damping ratio ζ 0.01Cubic nonlinear coefficient δ 0.5Quadratic nonlinear coefficient λ 0.5Quintic nonlinear coefficient β 0.5Time constant ratio α 0.05Coupling term κ 0.05Excitation intensity D 0.01

Fig. 3. Stationary probability density of the amplitude, the displacement. and the velocity of the Duffing-type VEH.

Fig. 4. Variation of the excitation intensity with the mean square voltage and the mean square power.

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283 271

maximal power. The result is in full agreement with those in references [22,27]. In all cases in Figs. 4–8, the analyticalsolutions are in good agreement well with the numerical ones. Therefore, the approximate analytical solution of stochasticaveraging method is supported by the Monte Carlo numerical simulations.

Fig. 5. Variation of the damping with the mean square voltage and the mean square power.

Fig. 6. Variation of the cubic nonlinearity with the mean square voltage and the mean square power.

Fig. 7. Variation of the coupling coefficient with the mean square voltage and the mean square power.

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283272

Fig. 8. Variation of the time constant ratio with the mean square voltage and the mean square power.

Fig. 9. Schematics of cubic-quintic Duffing-type VEH [34].

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283 273

7. Applied to cubic-quintic Duffing-type VEH

Cao et al. [34] designed a cubic-quintic Duffing-type VEH, investigated the influence of potential well depth on har-vesting performance under harmonic excitation (Fig. 9). The experiments and accompanying numerical simulations showthat the potential well depth will enhance the broadband performance. So far, to the authors’ best knowledge, there is nothe stochastic analysis on the cubic-quintic Duffing-type VEH under random excitation. To address the lacks of research inthis aspect, we consider the cubic-quintic Duffing-type VEH as an application of the proposed stochastic averaging method.

Adopting the same procedure as used in Section 3 gives the non-dimensional electromechanical coupling equations

€Xþ2ζ _XþXþδX3þβX5þκY ¼ ξðtÞ (42a)

_YþαY ¼ _X (42b)

The conservative system associated with Eq. (42a) has periodic solution of form (13). According to Eq. (6) and Section 4,we can obtain

f ðXÞ ¼ ω2ðA;ΦÞκα2þω2ðA;ΦÞþ1

�XþδX3þβX5;

V�ðXÞ ¼ 12

ω2ðA;ΦÞκα2þω2ðA;ΦÞþ1

�X2þ1

4δX4þ1

6βX6 (43)

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283274

Using Eq. (17), the frequency function equation can be obtained

ω4�ðκ�α2þΛÞω2�α2Λ¼ 0 (44)

where

Λ¼ δA2=2þβA4=3� �

ð1þ cos 2ΦÞþβA4=3 cos 4Φ:

and the corresponding to frequency function is

ωðA;ΦÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðκ�α2þΛÞþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðκ�α2þΛÞ2þ4α2Λ

q2

vuut(45)

ωðA;ΦÞ can be approximated by the finite Fourier series

ωðA;ΦÞ ¼ω0ðAÞþω1ðAÞ cos 2Φþω2ðAÞ cos 4Φ (46)

Average frequency approximately

ωðAÞ ¼ ω0ðAÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiκþ1þδA2þ2βA4=3

q(47)

Substitution Eqs. (13) and (15) into Eq. (42), applying the same procedure in Section 5 yields,

a1ðAÞ ¼Aðκþ1þδA2þ2βA4=3Þ2ðκþ1þδA2þβA4Þ

2ζþ ακ2

κþ1þδA2

�

�DβA3½2ðκþ1ÞþδA2�3ðκþ1þδA2þβA4Þ3

þDðκþ1þδA2þ2βA4=3Þ2Aðκþ1þδA2þβA4Þ2

;

a2ðAÞ ¼ 0; b11ðAÞ ¼Dðκþ1þδA2þ2βA4=3Þðκþ1þδA2þβA4Þ2

;

b12ðAÞ ¼ b21ðAÞ ¼ 0; b22ðAÞ ¼Dðκþ1þδA2þ2βA4=3Þ½ðκþ1ÞAþδA3þβA5�2

(48)

The stationary probability density of amplitude of Eq. (42) is

pðaÞ ¼ Cðkaþδa3þβa5Þexp �ξðka2þδa4=2þβa5=3Þ=ð2DÞ� �(49)

where C is a normalization constant as defined after Eq. (28). The stationary probability density of displacement and velocitycan be further obtained from p(H) as follows

pðx; _xÞ ¼ Cexp �ξð_x2þkx2þδx4=2þβx6=3Þ=ð2DÞh i

(50)

Using Eqs. (31), (32) and (50), one obtains the mean-square voltage and the power.The influences of the system parameters on the stationary mean square voltage and the output power will be reported.

For given parameters, the mean square voltage and the output power can be determined by Eqs. (31), (32) and (50),respectively. For specified parameters and initial conditions, the stationary probability density of the Eq. (42) can be cal-culated via the Monte-Carlo simulations. In the following, the solid lines are the analytical results of the proposed method,the solid circles are numerical results based on Eq. (42). The initial conditions are set as the static equilibrium positionXð0Þ ¼ 0; _Xð0Þ ¼ 0;Yð0Þ ¼ 0 and the system parameters listed in Table 1.

Fig. 10 shows the probability densities of the displacement, the velocity, and the amplitude determined by Eqs. (49) and(50) and those by Monte Carlo numerical simulations of Eq. (42). It is demonstrate that the two results are in full agreement.It is confirmed that the probability densities of the displacement and the velocity are Gaussian distribution and that of theamplitude is Rayleigh distribution.

The influences of the system parameters of the excitation intensity D, the damping coefficient ζ, the cubic nonlinearcoefficient δ, the quintic nonlinear coefficient β, the piezoelectric coupling coefficient κ in the mechanical equation, the timeconstant ratio α on the stationary mean square voltage and the output power are illustrated in Figs. 11–16. The mean squarevoltage and the output power increases with increasing the excitation intensity, dramatically decreases with the dampingcoefficient ζ and decreases proportionally with the cubic nonlinear coefficient δ and the quintic nonlinear coefficient β. It is alsoevident that the cubic nonlinearity and the quintic nonlinearity cannot increase the mean square voltage and the output power.

Fig. 15 illustrates that the mean square voltage decreases with the coupling coefficient increases, and there exists anoptimal piezoelectric coupling coefficient to obtain the maximal output power. Fig. 16 indicates that there exists an optimaltime constant ratio to obtain the maximal output power, which means that there exists an optimal resistance to generatethe maximal power. The consistency of the analytical results and the results from numerical simulation demonstrates theaccuracy of the stochastic averaging method.

Fig. 10. Stationary probability density of the amplitude, the displacement. and the velocity of the cubic-quintic Duffing-type VEH.

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283 275

8. Applied to quadratic-cubic VEH

The quadratic-cubic VEH is an asymmetric systemwhich includes quadratic, and cubic nonlinearity in the restoring force[20]. Examples of such VEHs include, but are not limited to, the asymmetrically magnetic force inductive harvester and thebistable VEH undergoes an intrawell motion which is expanded in a Taylor series around an equilibria point [18,35] (Fig. 17).

Adopting the same non-dimensional procedure as used in reference [19,20], one obtains the following non-dimensionalelectromechanical coupling equations

€Xþ2ζ _XþXþλX2þδX3þκY ¼ ξðtÞ (51a)

_YþαY ¼ _X (51b)

Using the same stochastic averaging and deterministic averaging method in Section 5, one has

a1ðAÞ ¼ �Aðκþ1þλAþδA2Þ2ðκþ1þλAþδA2Þ

2ζþ ακ

κþ1þδA2

� þ D½2ðκþ1ÞþλA�4Aðκþ1þλAþδA2Þ2

a2ðAÞ ¼ 0; b11ðAÞ ¼Dðκþ1þλAþδA2Þðκþ1þλAþδA2Þ2

;

b12ðAÞ ¼ b21ðAÞ ¼ 0; b22ðAÞ ¼Dðκþ1þλAþδA2ÞA2ðκþ1þλAþδA2Þ2

(52)

The stationary probability density of amplitude of Eq. (51) is

pðaÞ ¼ Cðkaþλa2þδa3Þexp �ξðka2þ2λa3=3þδa4=2Þ=ð2DÞ� �(53)

Fig. 12. Influence of the damping on the mean square voltage and the mean square power.

Fig. 13. Influence of the cubic nonlinearity on the mean square voltage and the mean square power.

Fig. 11. Influence of the excitation intensity on the mean square voltage and the mean square power.

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283276

Fig. 15. Influence of the coupling coefficient on the mean square voltage and the mean square power.

Fig. 16. Influence of the time constant ratio on the mean square voltage and the mean square power.

Fig. 14. Influence of the quintic nonlinearity on the mean square voltage and the mean square power.

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283 277

Fig. 17. Schematics of quadratic-cubic Duffing-type VEH [21,10,35].

Fig. 18. Stationary probability density of the amplitude, the displacement. and the velocity of the quadratic-cubic Duffing-type VEH.

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283278

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283 279

pðx; _xÞ ¼ Cexp �ξð_x2þkx2þ2λx3=3þδx4=2Þ=ð2DÞh i

(54)

where C is a normalization constant as defined after Eq. (28). Using Eqs. (31), (32) and (54), one obtains the mean-squarevoltage and the power.

To discuss the influences of the system parameters on the expected stationary output power, variations of the stationarymean square voltage and the output power will be presented for different tuning parameters. For specified parameters, themean square voltage and the output power can be determined by Eqs. (31), (32) and (54), respectively. For specifiedparameters and initial conditions, the stationary probability density of the Eqs. (51) can be calculated through the Monte-Carlo simulations. In the following, the solid lines are the analytical results, the solid circles are numerical results, the systemparameters listed in Table 1. The initial conditions are set as the static equilibrium position Xð0Þ ¼ 0; _Xð0Þ ¼ 0;Yð0Þ ¼ 0.

The stationary probability densities of the amplitude, the displacement, and the velocity are shown in Fig. 18. The Fig. 18clearly demonstrates that the results obtained by using the proposed stochastic averaging method are in excellent agree-ment with those from Monte Carlo numerical simulations. In addition, the Fig. 18 shows that the probability densities of thedisplacement and the velocity are Gaussian distribution and that of the amplitude is Rayleigh distribution.

Figs. 19–24 depict the variation of the mean square voltage and the output power with excitation intensity D, thedamping coefficientζ, the cubic nonlinear coefficient δ, the quadratic nonlinear coefficientλ, the piezoelectric couplingcoefficient in the mechanical equation κ, the time constant ratio α. The mean square voltage and the output power increaseswith the increasing D, decreases with the damping coefficient ζ and the cubic nonlinear coefficient δ. It can be seen fromFig. 21 that the mean square voltage and the output power increases with the increasing quadratic nonlinear coefficient.

Fig. 19. Variation of the excitation intensity on the mean square voltage and the mean square power.

Fig. 20. Variation of the damping on the mean square voltage and the mean square power.

Fig. 21. Variation of the quadratic nonlinearity on the mean square voltage and the mean square power.

Fig. 22. Variation of the cubic nonlinearity on the mean square voltage and the mean square power.

Fig. 23. Variation of the coupling coefficient on the mean square voltage and the mean square power.

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283280

Fig. 24. Variation of the time constant ratio on the mean square voltage and the mean square power.

Fig. 25. Variation of the pure quadratic nonlinearity on the mean square voltage and the mean square power.

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283 281

Therefore, the quadratic combined with properly cubic nonlinearities can increases the mean square voltage and the outputpower of vibration energy harvester under Gaussian white noise excitation.

Fig. 23 indicates that the mean square voltage decreases with the coupling coefficient increases, and there exists anoptimal piezoelectric coupling coefficient to obtain the maximal output power. Fig. 24 demonstrates that there exists anoptimal time constant ratio to obtain the maximal output power, which means that there exists an optimal resistance togenerate the maximal power. In all cases in Figs. 19–24, the analytical solutions are validated through Monte Carlo numericalsimulations.

The steady state response of an energy harvester with quadratic nonlinearity only is calculated. Fig. 25 plots the variationof the mean square voltage and the output power with the quadratic nonlinear coefficient. This indicates that the quadraticnonlinearity can increase the mean square voltage and the output power of the energy harvester under Gaussian whitenoise excitation.

9. Conclusions

This work focuses on the output power of energy harvesting systems under Gaussian white noise excitation. The gen-eralized harmonic transformation is performed to decouple the electromechanically coupled equations. The stochasticaveraging method is employed to derive the averaged Itô and the averaged FPK equations of the decoupled system. Theexact stationary solution of the associated averaged FPK equation is derived. The probability densities of the displacement,the velocity, the amplitude, and the power of the stationary response are obtained. The proposed stochastic averaging

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283282

procedure is applied to the Duffing-type VEH, cubic-quintic nonlinearity VEH and quadratic-cubic nonlinearity VEH. Theinfluences of the tuning parameters of the system on the mean square voltage and power are discussed. Numerical results ofsteady-state solutions are calculated by the Monte Carlo simulations. Excellent agreement is found between the results bythe proposed stochastic averaging procedure and the Monte Carlo simulations. It is concluded that the stochastic averagingmethod can treat the coupled electromechanical systems of energy harvesters under Gaussian white noise excitations.

The investigation yields the following conclusions: (1) the proposed stochastic averaging method under Gaussian whitenoise excitations can be applied to obtain the exact stationary probability densities of the averaged FPK equation; (2) theproposed stochastic averaging method can be used to determine the expected value of the mean square voltage and thepower under Gaussian white noise excitations; (3) the mean square voltage and the power increase dramatically with theexcitation intensity; (4) the mean square voltage and the power decrease with the damping coefficient, the cubic nonlinearcoefficient and the quintic nonlinear coefficient; (5) there exist an optimal piezoelectric coupling coefficient and an optimaltime constant ratio, respectively, to achieve the maximal output power; (6) the quadratic nonlinearity only and quadraticcombined with properly cubic nonlinearities can increase the mean square voltage and the output power of vibrationenergy harvester under Gaussian white noise excitation, respectively.

Acknowledgments

This work was supported by the State Key Program of National Natural Science Foundation of China (No. 11232009) andthe National Natural Science Foundation of China (No. 11572182).

References

[1] L.H. Tang, Y.W. Yang, C.K. Soh, Toward broadband vibration-based energy harvesting, Journal of Intelligent Material Systems and Structures 21 (2010)1867–1897.

[2] S.P. Pellegrini, N. Tolou, M. Schenk, J.L. Herder, Bistable vibration energy harvesters: a review, Journal of Intelligent Material Systems and Structures 24(2012) 1303–1312.

[3] R.L. Harne, K.W. Wang, A review of the recent research on vibration energy harvesting via bistable systems, Smart Materials and Structures 22 (2013)023001.

[4] M.F. Daqaq, R. Masana, A. Erturk, D.D. Quinn, On the role of nonlinearities in vibratory energy harvesting: a critical review and discussion, AppliedMechanics Reviews 66 (2014) 040801.

[5] A. Erturk, D.J. Inman, Piezoelectric energy harvesting, Wiley, New York, 2011.[6] N. Elvin, A. Erturk, Advances in energy harvesting methods, Springer, New York, 2013.[7] V. Challa, M. Prasad, Y. Shi, F. Fisher, A vibration energy harvesting device with bidirectional resonance frequency tenability, Smart Materials and

Structures 75 (2008) 015035.[8] S.M. Shahruz, Design of mechanical band-pass filters for energy scavenging, Journal of Sound and Vibration 292 (2006) 987–998.[9] B.P. Mann, N.D. Sims, Energy harvesting from the nonlinear oscillations of magnetic levitation, Journal of Sound and Vibration 319 (2009) 515–530.

[10] A. Erturk, D.J. Inman, Broadband piezoelectric power generation on high-energy orbits of the bistable Duffing oscillator with electromechanicalcoupling, Journal of Sound and Vibration 330 (2010) 2339–2353.

[11] Y. Zhu, J.W. Zu, Enhanced buckled-beam piezoelectric energy harvesting using midpoint magnetic force, Applied Physics Letters 103 (2013) 041905.[12] S. Zhou, J. Cao, A. Erturk, J. Lin, Enhanced broadband piezoelectric energy harvesting using rotatable magnets, Applied Physics Letters 102 (2013) 173901.[13] F. Cottone, H. Vocca, L. Gammaitoni, Nonlinear energy harvesting, Physics Review Letters 102 (2009) 080601.[14] M.F. Daqaq, Response of uni-modal Duffing-type harvesters to random forced excitations, Journal of Sound and Vibration 329 (2010) 3621–3631.[15] M.F. Daqaq, Transduction of a bistable inductive generator driven by white and exponentially correlated Gaussian noise, Journal of Sound and Vibration

330 (2011) 2554–2564.[16] P.L. Green, K. Worden, K. Atallah, N.D. Sims, The benefits of Duffing-type nonlinearities and electrical optimisation of a mono-stable energy harvester

under white Gaussian excitations, Journal of Sound and Vibration 331 (2012) 4504–4517.[17] S. Ali, F.S. Adhikari, M.I. Friswell, S. Narayanan, The analysis of piezomagnetoelastic energy harvesters under broadband random excitations, Journal of

Applied Physics 109 (2011) 074904.[18] R. Masana, M.F. Daqaq, Response of duffing-type harvesters to band-limited noise, Journal of Sound and Vibration 332 (2013) 6755–6767.[19] M.F. Daqaq, On intentional introduction of stiffness nonlinearities for energy harvesting under white Gaussian excitations, Nonlinear Dynamics 69

(2012) 1063–1079.[20] Q.F. He, M.F. Daqaq, Influence of potential function asymmetries on the performance of nonlinear energy harvesters under white noise, Journal of

Sound and Vibration 33 (2014) 3479–3489.[21] Q.F. He, M.F. Daqaq, New insights into utilizing bistability for energy harvesting under white noise, Journal of Vibration and Acoustics 137 (2015) 021009.[22] M. Xu, X.L. Jin, Y. Wang, Z.L. Huang, Stochastic averaging for nonlinear vibration energy harvesting system, Nonlinear Dynamics 78 (2014) 1451–1459.[23] P. Kumar, S. Narayanan, S. Adhikari, M.I. Friswell, Fokker–Planck equation analysis of randomly excited nonlinear energy harvester, Journal of Sound

and Vibration 333 (2014) 2040–2053.[24] X.L. Jin, Y. Wang, M. Xu, Z.L. Huang, Semi-analytical solution of random response for nonlinear vibration energy harvesters, Journal of Sound and

Vibration 340 (2015) 267–282.[25] J.B. Roberts, Response of nonlinear mechanical systems to random excitation, Part 1: Markov methods, The Shock and Vibration Digest 13 (1981) 17–28.[26] S.H. Crandall, W.Q. Zhu, Random vibration: a survey of recent developments, ASME Journal of Applied Mechanics 50 (1983) 935–962.[27] W.Q. Zhu, Recent developments and applications of stochastic averaging method in random vibration, Applied Mechanics Reviews 49 (1996) 72–80.[28] J.B. Roberts, Effect of parametric excitation on ship rolling motion in random waves, Journal of Ship Research 26 (1982) 246–253.[29] J.B. Roberts, P.D. Spanos, Stochastic averaging: an approximate method of solving random vibration problems, International Journal of Non-Linear

Mechanics 21 (1986) 111–134.[30] Z.L. Huang, W.Q. Zhu, Y. Suzuki, Stochastic averaging of strongly non-linear oscillators under combined harmonic and white-noise excitations, Journal

of Sound and vibration 238 (2000) 233–256.[31] W.Q. Zhu, Z.L. Huang, Y. Suzuki, Response and stability of strongly non-linear oscillators under wide-band random excitation, International Journal of

Non-Linear Mechanics 36 (2001) 1235–1250.

W.-A. Jiang, L.-Q. Chen / Journal of Sound and Vibration 377 (2016) 264–283 283

[32] Z. Xu, Y.K. Cheung, Averaging method using generalized harmonic functions for strongly nonlinear oscillators, Journal of Sound and vibration 174 (1994)563–576.

[33] R.Z. Khasminskii, On the principle of averaging for the Itô stochastic differential equation, Kybernetika 3 (1968) 260–279.[34] J.Y. Cao, S.X. Zhou, W. Wang, J. Lin, Influence of potential well depth on nonlinear tristable energy harvesting, Applied Physics Letters 106 (2015) 173903.[35] W.A. Jiang, L.Q. Chen, Snap-through piezoelectric energy harvesting, Journal of Sound and Vibration 333 (2014) 4314–4325.