Probabilistic response analysis of nonlinear vibration energy harvesting system driven by Gaussian colored noise

Chaos, Solitons and Fractals 104 (2017) 806–812

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Probabilistic response analysis of nonlinear vibration energy harvesting system driven by Gaussian colored noise Di Liu a,∗, Yong Xu b,d,e, Junlin Li c a

School of Mathematics, Shanxi University, Taiyuan, 030006, PR China Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, PR China c School of Applied Science, Taiyuan University of Science and Technology, Taiyuan, 030024, PR China d School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 750021, PR China e Potsdam Institute for Climate Impact Research, Potsdam 14412, Germany b

a r t i c l e

i n f o

Article history: Received 20 February 2017 Accepted 18 September 2017

Keywords: Nonlinear vibration energy harvesting Quasi-conservative averaging method Mean-square electric voltage Gaussian colored noise Correlation time

a b s t r a c t A new quasi-conservative stochastic averaging method is proposed to analyze the Probabilistic response of nonlinear vibration energy harvesting (VEH) system driven by exponentially correlated Gaussian colored noise. By introducing a method combining a transformation and the residual phase, the nonlinear vibration electromechanical coupling system is equivalent to a single degree of freedom system, which contains the energy-dependent frequency functions. Then the corresponding drift and diffusion coefficients of the averaged Itoˆ stochastic differential equation for the equivalent nonlinear system are derived, which are dependent on the correlation time of Gaussian colored noise. The probability density function (PDF) of stationary responses is derived through solving the associated Fokker–Plank–Kolmogorov (FPK) equation. Finally, the mean-square electric voltage and mean output power are analytically obtained through the relation between the electric voltage and the vibration displacement, and the output power has a linear square relationship with the electric voltage, respectively. The main results on probabilistic response of VEH system are obtained to illustrate the proposed stochastic averaging method, and Monte Carlo (MC) simulation method is also conducted to show that the proposed method is quite effective. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Vibration energy harvesting (VEH) is the process which energy is derived from external sources, captured and stored for small, wireless autonomous devices, like those used in wearable electronics and wireless sensor networks [1]. The basic operation principles of those devices are what convert available mechanical energy into electrical energy through the electromagnetic, piezoelectric, and electrostatic transduction mechanisms. Among them, the piezoelectric energy harvester plays a great role with its advantages in inducing excellent power densities, relatively high voltages, and low currents, and has already been widely used in engineering field. Since the energy source for Vibration energy harvesters is present as ambient background and is free, and therefore the research of the dynamical behavior of VEH system has attracted more and more attention over the past 10 years. At the start of

∗

Corresponding author. E-mail addresses: [email protected] (D. Liu), [email protected] (Y. Xu).

https://doi.org/10.1016/j.chaos.2017.09.027 0960-0779/© 2017 Elsevier Ltd. All rights reserved.

the study, most piezoelectric energy harvesters adopted the linear resonant model[2–4]. However, this model has a very narrow frequency bandwidth, which will result in the system become incapable of harvesting energy efficiently when the excitation frequency over a relatively wide range of the harvesters fundamental frequency. To remedy this problem, the unique advantages of nonlinearity of restoring force of the system had been suggested and applied. Comparing to the linear model, nonlinear model has a wider steady-state frequency bandwidth [5–7]. Among them, the mono-stable, bi-stable and tri-stable nonlinear oscillator, as the simplest nonlinearity model, had been used in the design of many VEH devices, with the addition of electromechanical coupling for the harvesting circuit. For instance, Triplett and Quinn [8] used the Poincare–Lindstedt perturbation method to analyze the response of a nonlinear lumped-model by including a nonlinear term in the electromechanical coefficient, and found that the nonlinearities in the electromechanical coupling can increase the harvested electrical power. Mann and Owens [9] used the magnetic interactions to create an electromagnetic VEH system of nonlinear bi-stable potential well and to validate the potential well escape phenomenon, and also testified the phenomenon will broaden the

D. Liu et al. / Chaos, Solitons and Fractals 104 (2017) 806–812

frequency response through the theory and experiments. Stanton et al. [10] and Zhou et al. [11] investigated the dynamic responses of the bi-stable and tri-stable energy harvester by means of the harmonic balance method, respectively. There is no doubt that the randomness exists in most realworld circumstances widely and therefore the nonlinear VEH, especially in stationary and non-stationary stochastic vibratory environments have attracted growing interest among researchers [12–16]. Specially, to better understand the effect of stochastic excitation on energy harvesting efficiency, some effective techniques had been proposed to investigate the stochastic response of VEH system with stochastic base acceleration. Jiang and Chen [17] used the Van Kampen expansion method to discuss the effects of the system parameters on the stochastic response of the piezoelectric coupling system, and recently they also presented the stochastic averaging method to analyze the response of nonlinear VEH system subject to external Gaussian white noise[18,19]. He and Daqaq [20,21] illustrated the effects of the potential energy function on the mean output power through the statistical linearization techniques and the steady-state approximation. Xu et al. [22] proposed a stochastic method combined with generalized harmonic transformation to analytically evaluate the mean-square electric voltage and mean output power for the nonlinear VEH system driven by Gaussian white noise. Recently, Fokou et al. [23] investigated the response, the stability and the reliability of a sandwiched piezoelectric buckled beam with axial compressive force under Gaussian white noise by means of stochastic averaging method. Most studies of VEH systems have considered either sinusoidal excitations or Gaussian white noise excitations [24–28]. However, actual environmental excitations in applications can deviate from these idealizations. For this reason, the Ornstein–Uhlenbeck process, i.e., external colored noise had been attracting a lot of attention [29,30]. For example, Daqaq [31] employed the decouple approximate FPK equation methods to obtain the mean output power of a Duffing oscillator driven by exponentially correlated Gaussian noise. Mendez et al. [32] discussed the performance of a linear electromechanical energy harvesting system subjected to the arbitrary colored noise. Recently, a linear electromechanical oscillator with a random ambient excitation was considered as a VEH model, which had been researched by Bobryk and Yurchenko [33], they found that a parametric colored excitation can have a dramatic effect on the enhancement of the energy harvesting. However, there is a lock of effective analytical method to research the probabilistic response of nonlinear VEH system under the Gaussian colored noise excitation, and then to understand the effect of stochastic excitation on energy harvesting efficiency. As mentioned above, our aims are to provide a stochastic approximate method to derive the probabilistic response and mean output power of nonlinear VEH system, and then to determine the effect of Gaussian colored noise on the energy harvesting efficiency. The paper is organized as follows. We first present the basic mathematical model of our consideration stochastic VEH system with Gaussian colored noise, and this stochastic model will be equivalent to a single degree of freedom system containing the energy-dependent frequency function through a method, which combines a transformation and the residual phase. In Section 3, the probabilistic response and the mean output power of nonlinear VEH system are derived under the assumption that the nonlinear VEH system is quasi-conservative. Results obtained from the proposed procedure are verified by direct numerical simulation results in Section 4. Furthermore, the effects of correlation time, the random excitations density, viscous damping coefficient and nonlinear stiffness coefficient on mean output power of the nonlinear VEH system is also investigated. A conclusion is given in the last section.

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2. Nonlinear vibration energy harvesting system with Gaussian colored noise The mathematical model in this investigation is considered to represent the dynamics of a class of piezoelectric vibratory energy harvester, which can be presented by a base-excited spring-massdamper system coupled to a capacitive energy harvesting circuit, as shown in Fig. 1. The equation of motion can be described as [22,31]

mX¨¯ + cX¯˙ + K1 X¯ + K3 X¯ 3 + θ V¯ = −mX¨¯c (τ ),

C pV¯˙ +

(1a)

1 ¯˙ V¯ = θ X, R

(1b)

where X¯ is the relative displacement of an inertial mass m. c, K1 and K3 denote the linear viscous damping coefficient, the linear, and nonlinear stiffness coefficients of VEH system, respectively. θ represents a linear electromechanical coupling coefficient. V is the voltage measured across an equivalent resistive load R, and Cp is the piezoelectric capacitance. X¨¯c (τ ) is the stochastic base acceleration, and the dot represents the derivative with respect to time τ . In this investigative, we consider a stochastic excitation source as Gaussian colored noise, which is a Gaussian noise with exponential correlated time. Applying the appropriate rescaling, one can obtain the following dimensionless model for a couple of nonlinear VEH system:

X¨ + cX˙ + ω02 X + γ X 3 + β V = ξ (t ),

(2a)

V˙ + λV = X˙ ,

(2b)

where the dot represents differentiation with respect to t. The stochastic base acceleration excitation ξ (t) is the Gaussian colored noise, which has the following statistical properties

E [ξ (t )] = 0,

R(τ ) = E [ξ (t )ξ (t + τ )] =

D1

τ1



exp −

 |τ | . (3) τ1

Integrating the electric Eq. (2b), the following approximation expression of electric voltage can be written as



. V (t ) =

t

0

X˙ (s ) exp [−λ(t − s )]ds =



t 0

X˙ (t − τ ) exp (−λτ )dτ . (4)

By introducing a transformation

sgnX



√ H cos θ , √ X˙ = − 2H sin θ ,

U (X ) =

(5)

where U(X) and H(t) are the potential energy and the total energy for undamped free motion, which cannot be explicitly expressed in t this stage. θ = θ (t ) = 0 ω (H )ds + φ (t ) is the total phase, φ (t) denotes the residual phase which is slowly varying, and ω(H) denote energy-dependent frequency function. Therefore, the following approximations will be obtained by φ (t) is slowly varying

θ (t − τ ) =



t−τ

ω (H )dτ + φ (t − τ ) ≈ θ (t ) − ω (H )τ , √ X˙ (t − τ ) = − 2H sin θ (t − τ ) 0

= X˙ cos (ω (H )τ ) + sgnX



2U (X ) sin (ω (H )τ ).

(6)

Substituting Eq. (6) into Eq. (4), the nonlinear relationship between electric voltage and state variables can be written as

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D. Liu et al. / Chaos, Solitons and Fractals 104 (2017) 806–812

Fig. 1. A simplified representation of a piezoelectric energy harvester.

 ω (H ) λ sgnX 2U (X ) + 2 X˙ 2 λ + ω (H ) λ + ω 2 (H ) ω 2 (H ) λ = 2 X+ 2 X˙ . λ + ω 2 (H ) λ + ω 2 (H )

V (t ) =

two first-order differential equation for H(t) and φ (t) as follows

2

 ¯ 2 (H ) X + γ X 3 = ξ (t ), X¨ + (c + C (H ) )X˙ + ω02 + ω

(8)

where C (H ) = βλ/[λ2 + ω2 (H )] and ω ¯ 2 (H ) = βω2 (H )/[λ2 + ω2 (H )] are the equivalent damping coefficient and the equivalent stiffness coefficient of electric voltage of the nonlinear piezoelectric VEH system, respectively. The potential energy U(X) and the total energy H(t), associated with the undamped free vibration response of the equivalent nonlinear system may be defined by the relation

 1 2 1 ω0 + ω¯ 2 (H ) X 2 + γ X 4 , 2 4 1 2 ˙ H = X + U (X ). 2

(11a)

φ˙ = f2 (H, θ ) + g2 (H, θ )ξ (t ),

(11b)

(7)

Then the nonlinear coupled system of Eqs. (2a) and (2b) are approximately replaced by the following equivalent nonlinear system:



H˙ = f1 (H, θ ) + g1 (H, θ )ξ (t ),

U (X ) =

(9)

where

f1 (H, θ ) = −2H (c + C (H ) ) sin

2



ω (H ) =

AH −AH



1 2H − 2U (X )

dH = mτ1 (t, H )dt + στ1 (t, H )dB(t ),



f1 (H, θ ) +

mτ1 (t, H ) =

+g2 (t + τ )



0 −∞



g1 (t + τ )

∂ g1 (t ) R ( τ )d τ ∂φ

= −(c + C (H ) )u2 (H ) + (10)

where AH denotes the max displacement determined by H = U (AH ), which depends on the energy level H. Fig. 2(a), (b) and (c) show the total energy function H, energydependent amplitude AH and frequency ω(H) curve for the state variables of unperturbed system, respectively.

(13)

where B(t) is the standard Browian motion, mτ1 (t, H ) and στ1 (t, H ) are averaged drift and diffusion coefficients of Markov processes, which should be of the following forms:

dX,

2π , TH

(12)

Due to the energy process H is approximate Markovian, and the quasi-conservative averaging procedure can be applied to obtain the averaged Itoˆ equation for H(t) as follow

The period and frequency of undamped free vibration can be expressed as

TH = 2

θ,

f2 (H, θ ) = −(c + C (H ) ) sin θ cos θ , √ 1 g1 (H, θ ) = − 2H sin θ , g2 (H, θ ) = − √ cos θ . 2H

στ1 (t, H ) =



2

∞

−∞

∂ g1 (t ) ∂H 

(14)

t

D1 1 + τ12 ω2 (H )

 g1 (t + τ )g1 (t )R(τ )dτ

=2 t

,

(14)

D1 u2 (H ), 1 + τ12 ω2 (H ) (15)

in which u2 (H) is a energy-dependent function given by

 

u2 (H ) = X˙ 2 = t

4 TH



0

AH

X˙ dX =

4 TH



AH 0



2H − 2U (X )dX.

(16)

3. Probabilistic response

3.2. The probabilistic response statistics of the VEH system

3.1. Stochastic averaging of quasi-conservative

The averaged FPK equation associated with Itoˆ Eq. (13) is of the form

The equivalent nonlinear system in Eq. (8) can be investigated through the quasi-conservative stochastic averaging technique for the case with small damping coefficient c + C (H ) = O( ) and

small excitation intensity D1 = O  1/2 . Therefore, the energy input by external excitation and the energy dissipated by the total damping is quite smaller than the total energy. The total energy H are slowly varying comparing to the system displacement and velocity. Substituting Eq. (5) into Eq. (8), one obtains the following

  ∂ pτ1 ∂ [mτ1 (H ) pτ1 ] 1 ∂ 2 στ21 (H ) pτ1 =− + , ∂t ∂H 2 ∂ H2

(17)

where pτ1 = pτ1 (t, H |H0 ) is the transition probability density of H depended on the correlation time of Gaussian colored noise with initial condition

pτ1 (t, H |H0 ) = δ (H − H0 ).

(18)

D. Liu et al. / Chaos, Solitons and Fractals 104 (2017) 806–812

809

(a) 25 20 15 H

2

10 1

5 0 6

0 4

2

−1

0

−2

−4

−6

dX/dt

(b) 2.8

X

−2

(c) 3

2.5

2

1.9

AH

ω (H)

2.2 2

1.6

2 1

1.3

1

1

1 6

0 6

0

4 2 0

−1

0 4

2

−2 −4 −6

dX/dt

−1

0

−2

X

−2

dX/dt

−4

−6

−2

X

Fig. 2. The energy, energy-dependent frequency and the max displacement cure for the state variables of unperturbed system.

Then the stationary probability density of the total energy can be obtained analytically

pτ1 (H ) =



C0 exp 2 στ21 (H )

H

0

mτ1 (τ1 , u ) du , στ21 (u )

(19)

where C0 is a normalization constant. The joint probability density of the displacement and the velocity can be obtained from pτ1 (H ) as follows





pτ1 X, X˙ =



pτ1 (H )  .  X˙ TH H= 2 +U (X )

(20)



Then the marginal probability density functions pτ1 (X ) and pτ1 X˙ can be obtained as follows:

pτ1 (X ) =



∞ −∞





pτ1 X, X˙ dX˙ ,



pτ1 X˙ =



∞ −∞





pτ1 X, X˙ dX.

(21)

The mean-square value of the electric voltage is then derived through the approximate relation in Eq. (7) of the electric voltage

and state variables as



2

E Vτ1





 2  2  ω 2 (H ) λ =E X +E X˙ λ2 + ω2 (H ) λ2 + ω2 (H )   λω2 (H ) + 2E

2 X X˙ λ2 + ω2 (H )  2

 ∞ ∞

ω2 X, X˙

X pτ1 X, X˙ dX dX˙ = 2 2 ˙ λ + ω X, X −∞ −∞  2  ∞ ∞

λ

X˙ pτ1 X, X˙ dX dX˙ + 2 2 ˙ λ + ω X, X −∞ −∞

 ∞ ∞

λω2 X, X˙ +2 

2 X X˙ pτ1 X, X˙ dX dX˙ . −∞ −∞ λ2 + ω 2 X, X˙

(22)

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D. Liu et al. / Chaos, Solitons and Fractals 104 (2017) 806–812

Fig. 3. The stationary probability densities of system displacement and velocity for different correlation times. Solid lines the present results; symbols results from MC simulation.





Fig. 4. The dependence of the mean-square electric voltage E Vτ21 on the excitation intensity 2D1 and correlation times τ 1 . Solid lines are the present results; symbols result from MC simulation.

Finally, according to the power has a linear square relationship with the electric voltage, the mean output power as



E [Pτ1 ] = λβ E Vτ21



(23)

4. Stochastic analysis of response In this section, results obtained by the extend the quasiconservative averaging procedure is shown and compared with those from direct numerical simulation to verify the effectiveness of the proposed method. The system parameters are set as c = 0.02, ω0 = 1, γ = 1.0, β = 0.5, λ = 0.05, D1 = 0.02, unless otherwise mentioned. The stationary marginal probability density

pτ1 (X ) and pτ1 X˙ of system displacement and system velocity are shown in Fig. 3, in which the solid line represents the analytical results obtained by the extend the quasi-conservative averaging

method while original point, square, rhombus and upper triangular denote the direct numerical simulation through Monte Carlo (MC) method for four different correlation times of Gaussian colored noise. It can be seen that they are in good agreement. Through analyzing the variation of Fig. 3 for different correlation time of Gaussian colored noise, we found that the increasing of correlation time will lead that the range of vibrating radius gets smaller of possibility for nonlinear VEH system. This indicates the correlation time of Gaussian colored noise will decrease the range of possible vibrating radius, and the has an essential affect on mean output power of the VEH system. To better understand the affect of correlation time of the noise on mean output power of the nonlinear VEH system, the changes of mean-square electric voltage with the density D1 of Gaussian colored noise, viscous damping coefficient c and nonlinear stiffness

D. Liu et al. / Chaos, Solitons and Fractals 104 (2017) 806–812









811

Fig. 5. The dependence of the mean-square electric voltage E Vτ21 on the damping coefficient c and the correlation times τ 1 . Solid lines the present results; symbols results from MC simulation.

Fig. 6. The dependence of the mean-square electric voltage E Vτ21 on the nonlinear stiffness coefficients γ and correlation times τ 1 . Solid lines the present results; symbols results from MC simulation.

coefficient γ for different correlation time are shown in Figs. 4–6, respectively. In Fig. 4, the mean-square electric voltage almost increases proportionally with the excitation intensity at any correlation time. It can be seen from Fig. 4(a), the noise intensity will increase the mean-square electric voltage in the case of arbitrary correlation time, but the increasing range reduces with the value of τ 1 increasing, that is the correlation time of colored noise will cause the drop of mean-square electric voltage. This means that the increase of correlation time of colored noise will hinder the production of output power in the same intensity. In Fig. 5(a) and (b), we studied the impact of a viscous damping coefficient c upon the mean-square electric voltage E(V2 ) under the different correlation time of Gaussian colored noise, and demonstrates our conclusions in some particular values through direct MC simulation. One can observe in Fig. 5(a), the mean-

square electric voltage become quickly low with the increase of correlation time for the smaller damping coefficient. However, the contribution of correlation time of the noise of the mean-square electric voltage is greatly reducing as the damping coefficient is increasing. Finally, the influence of the nonlinear stiffness coefficient and correlation time on the mean-square electric voltage is also illustrated in Fig. 6. The mean-square electric voltage almost linearly decreases with the increase of the nonlinear stiffness coefficients for any correlation time of Gaussian colored noise, and the contribution of the correlation time of the noise of the mean-square electric voltage has not change significantly as the nonlinear stiffness coefficient is increasing. Applying the mean output power relationship with the mean-square electric voltage, the correlation time of Gaussian colored noise will substantially degrade the performance of the nonlinear VEH system. Those

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results about the affect of the correlation time of Gaussian colored noise are consistent with an approximate solution of the FPK equation based on the decoupling theory [31]. 5. Conclusions In this manuscript, a stochastic method is developed combining both a transformation and the residual phase procedure to analyze the probabilistic response of nonlinear VEH system driven by exponentially correlated Gaussian colored noise. Firstly, the nonlinear vibration electromechanical coupling system can be approximated by an equivalent single degree of freedom system, which contains the energy-dependent frequency functions by means of the transformation mentioned previously. Secondly, under the assumption that the equivalent single degree of freedom system is quasi-conservative, the stochastic averaging method is used to derive the drift and diffusion coefficients of the averaged Itoˆ stochastic differential equation, which is dependent on the correlation time of Gaussian colored noise. Thirdly, the PDF of system displacement and velocity, and the joint PDF of them will be obtained through solving the corresponding FPK equation, and then the mean-square electric voltage is also derived through the relation between the electric voltage and the mechanical states. Finally, the effects of the excitation intensity, viscous damping coefficient and the nonlinear stiffness coefficients in the different correlation time of Gaussian colored noise are researched, we find that the correlation time will reduce the mean-square electric voltage when the other parameters unaltered. Applying the output power has a linear square relationship with the electric voltage, we can get such a conclusion: The correlation time of Gaussian colored noise also has an essential effect on mean output power of the nonlinear VEH system, and the corresponding results are verified through MC simulation in this paper. Acknowledgments This work is supported by the National Nature Science Foundation of China (No. 11402139, 11572247) and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (No. 2016114). References [1] Erturk A, Inman DJ. Piezoelectric energy harvesting. John Wiley & Sons; 2011. [2] Roundy S, Leland ES, Baker J, Carleton E, Reilly E, Lai E, et al. Improving power output for vibration-based energy scavengers. IEEE Pervasive Comput 2005;4(1):28–36. [3] Saha CR, O’Donnell T, Loder H, Beeby S, Tudor J. Optimization of an electromagnetic energy harvesting device. IEEE Trans Magn 2006;42(10):3509–11. [4] Stephen N. On energy harvesting from ambient vibration. J Sound Vib 2006;293(1):409–25. [5] Mann B, Sims N. Energy harvesting from the nonlinear oscillations of magnetic levitation. J Sound Vib 2009;319(1):515–30.

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