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Enhancing energy harvesting by a linear stochastic oscillatorBobryk, R. V.; Yurchenko, Daniil

Published in:Probabilistic Engineering Mechanics

DOI:10.1016/j.probengmech.2015.10.007

Publication date:2016

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Citation for published version (APA):Bobryk, R. V., & Yurchenko, D. (2016). Enhancing energy harvesting by a linear stochastic oscillator.Probabilistic Engineering Mechanics, 43, 1-4. DOI: 10.1016/j.probengmech.2015.10.007

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Author’s Accepted Manuscript

Enhancing energy harvesting by a linear stochasticoscillator

R.V. Bobryk, D. Yurchenko

PII: S0266-8920(15)30053-9DOI: http://dx.doi.org/10.1016/j.probengmech.2015.10.007Reference: PREM2873

To appear in: Probabilistic Engineering Mechanics

Received date: 13 June 2015Revised date: 16 September 2015Accepted date: 12 October 2015

Cite this article as: R.V. Bobryk and D. Yurchenko, Enhancing energy harvestingby a linear stochastic oscillator, Probabilistic Engineering Mechanics,http://dx.doi.org/10.1016/j.probengmech.2015.10.007

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www.elsevier.com/locate/probengmech

Enhancing energy harvesting by a linear

stochastic oscillator

R. V. Bobryk1

Institute of Mathematics, Jan Kochanowski University, Kielce, Poland

D. Yurchenko

Institute of Mechanical, Process and Energy Engineering, Heriot-Watt University,Edinburgh, UK

Abstract

A mathematical model of a linear electromechanical oscillator with a ran-domly fluctuating damping parameter as an energy harvester is considered.To demonstrate the idea and present an explicit analytical solution the varia-tions of the damping are taken in a form of a telegraphic process. It is shownthat substantial enhancement of the harvested energy can be reached if theproposed process has specially selected characteristics. A close relationshipwith the mean square stability of the oscillator is established. An analyticalresult for stationary electrical net mean power is presented.

1. Introduction

Energy harvesting is a conversion of ambient energy presented in the en-vironment into electrical energy that can be used or stored. The energyharvesters can be used as a replacement for batteries in low-power wirelesselectronic devices (e.g. sensor networks used in structural health monitoring).Among the power harvesting methods, the conversion of mechanical energyfrom vibrations into electrical energy via the piezoelectric effect is particu-larly effective and has received a great attention over the last ten years (seee.g. books [1, 2] and reviews [3–5]. Vibration-based energy harvesters are

Email addresses: Email: bobryk@ujk.edu.pl (R. V. Bobryk),Email:d.yurchenko@hw.ac.uk (D. Yurchenko)

Preprint submitted to Elsevier September 15, 2015

typically implemented as linear mechanical resonators based on the followingwell-known phenomenon which is usually described at undergraduate textsin differential equations.Consider the ubiquitous harmonic oscillator

x+ γx+ ω2x = η(t), (1)

where γ > 0 is a damping parameter, ω is a natural frequency, the excita-tion η(t) is harmonic, η(t) = a sin(Ωt). Then the solutions of Eq. (1) growdramatically if ω ≈ Ω and γ is small (the resonance phenomenon). A cruciallimitation of the mechanical resonators is that the best performance of thedevices is limited to a very narrow bandwidth around the natural frequency.If the excitation frequency deviates slightly from the resonance condition,the power output is drastically reduced, rendering linear resonant harvestersunsuitable for most practical applications. In a lot of cases the excitationexhibits random rather than deterministic behavior with a broad-band fre-quency spectrum.

When a broad-band external excitations is modeled by a zero-mean Gaus-sian white noise with intensity D, the power gained by the system is propor-tional to the noise intensity Pw = D. The latter is a very interesting result,since the harvested energy of a linear system is independent of the system’sparameters. This fact has been extended to the case of nonlinear systems [6]and it has been shown that the mass of the system and the noise intensityare the two factors influencing the harvesting capabilities of any dynamicalsystem, except the bistable systems, for instance [7–10]. It should be stressedthat the above result is the upper boundary for the considered class of sys-tems which may or may not be reached, whereas in the real life environment,where the excitation is rather broad band with finite power, any system hasto be adjusted for its best performance. For instance if one considers a linearsystem (1) under the Ornstein-Uhlenbeck process then power of (1) can beobtained exactly analytically using the method of moments:

Pµ =D(1 + γµ)

(1 + γµ+ ω2µ2= Pw

1

1 + ω2µ2

1+γµ

where D,µ -are parameters of the Ornstein-Uhlenbeck process. The aboveformula clearly shows that the harvested power depends on both the dampingcoefficient and natural frequency of the system.

2

In this paper we propose a promising approach for the vibration-based en-ergy harvesting staying within the linear mechanical system framework.Thevibratory system has a random time-varying variations of the damping pa-rameter with specially selected characteristics that allow significantly en-hance the energy harvesting. It can be achieved by using a magnetorheolog-ical (MR) damper [11]. The major idea behind the MR damper is the MRfluid which can change its viscosity under the influence of a magnetic field.It has been reported that MR dampers required relatively low amount of en-ergy, outperform passive devices and SMAs, as well as have a fast response[12–15].

2. Mathematical model of the energy harvester

The vibration-based energy harvester comprises two parts. The key partis a mechanical oscillator but the second one is a capacitor that stores theelectrical energy and is coupled with the oscillator by a transducer mechanismbased on the piezoelectricity. A simplified representation of the piezoelectricvibration - based energy harvester is shown in Fig. 1 where the left partis the capacitive energy harvesting circuit (capacitor) but the right one isthe base accelerated mechanical oscillator. Using Newton’s second law andOhm’s law one can obtain for the mechanical states and electric voltage thefollowing coupled equations [16]:

md2x(τ)

dτ 2+ b(τ)

dx(τ)

dτ+ kx(τ) + θV (τ) = −m

d2Z(τ)

dτ 2, (2)

RCpdV (τ)

dτ+ V (τ) = Rθ

dx(τ)

dτ, (3)

where x is a deflection of mass m, b(τ) is a time-varying damping coefficient,k is a stiffness coefficient, θ is a linear electromechanical coupling, Cp is acapacitance of the piezoelectric element, V is an electric voltage measured

across the equivalent resistive load R, and d2Z(τ)dτ2

is a base acceleration. Wesuppose that the damping coefficient has form

b(τ) = a[1 + σζ(ω1τ)],

where ζ(ω1τ) is some narrow band radnom variations of the constant dampingcoefficient a with a frequency ω1 and noise intensity 0 ≤ σ < 1. Introducethe substitutions

3

ω =

√k

m, t = ωτ, γ =

a√mk

, k1 =θ2

kCp

, k2 =1

RCpω,

x1 =x

lc, Z =

Z

lc, x3 =

V Cp

θlc

x2 =dx1

dt, η = −d2Z

dt2, ξ(t) = ζ

(ω1t

ω

),

(4)

where lc is a length scale. Then Eqs. (2), (3) are reduced to the followingsystem in the non-dimensional form:

dx1

dt= x2,

dx2

dt= −x1 − γ[1 + σξ(t)]x2 − k1x3 + η(t),

dx3

dt= x2 − k2x3,

(5)

We suppose that the process η(t) is a Gaussian white noise with inten-sity D. This process is considered as a good model of a broadband ambientexcitation (see e.g. [16–19]). The random process ξ(t) is a zero-mean tele-graphic process with the state −1, 1 and the transition rate α/2. Thisprocess is an ergodic Markov process and it is often used in applications (seee.g. books [20, 21]). The spectral characteristics of system (5) can be ob-tained exactly using the measuing filters approach [22]. It should be stressedthat such a choice of the process was dictated by the ability of deriving anexact analytical solution to set of equations (5) rather than anything else,thereby illustrating clearly the influence of parameters on the energy harvest-ing process. It is possible to consider any other narrow band exponentiallycorrelated process excitation process, but an explicit analytical solution maynot be available resulting in a lack of understanding the phenomenon behindthe proposed idea. The processes η(t) and ξ(t) are supposed independent.System (4) can be presented as a linear system of Ito stochastic differentialequations

dx1 = x2dt,

dx2 = −x1dt− γ[1 + σξ(t)]x2dt− k1x3dt+Ddw(t),

dx3 = x2dt− k2x3dt,

(6)

where w(t) is a standard Wiener process.

4

3. Analytical development and results

First consider the system (4) with η ≡ 0. The trivial solution of thissystem is called mean square asymptotically stable if all second moments ofany solution of the system tends to zero if t → ∞.

Let

α1 = α + γ

a0 = α1(1 + α(1 + α1 + k1) + [α1(2α + γ) + k1]k2+

+ α1k22 × [γ2k2 + k1k2 + γ(1 + k1 + k2

2)][8k2 + 4α(1 + k1)+

+ α(α + 2γ)(α + 2k2)],

a1 = b0 + b1k2 + b2k22 + b3k

32 + b4k

42,

b0 = 2α(1 + k1)[α4 + 2α3γ + 3αγ(1 + k1) + 2(1 + k1)

2 + α2(3 + γ2 + 3k1)],

b1 = 3α5γ + 8(1 + k1)2 + α4(8 + 7γ2 + 4k1) + α3γ × (24 + 4γ2 + 17k1)+

+ 4αγ(5 + 8k1 + 3k21) + 2α2(10 + 13k1 + 3k2

1 + 8γ2 + 7γ2k1),

b2 = 2α5 + 15α4γ + 16γ(1 + k1) + 4α2γ(13 + 3γ2 + 5k1) + 8α[3 + 2k1+

+ 2γ2(2 + k1)] + α3[25γ2 + 4(4 + k1)],

b3 = 2[4α4 + 13α3γ + 8(1 + γ2) + 4αγ(6 + γ2 + k1) + α2 × (13γ2 + 12 + 3k1)],

b4 = 2[5α3 + 8γ + 9α2γ + 2α(5 + 2γ2 + k1)],

a2 = 2k2(α+ 2k2)[α3 + 2k2 + α2(γ + k2) + α(2 + k1 + γk2)].

(7)It should be emphasized that the damping coefficient γ is always positive

because of our original assumption for 0 < σ < 1.Proposition. The trivial solution of system (4) with η ≡ 0 is mean

square asymptotically stable iff

σ < σ∗ =

√a1 −

√a21 − 4a0a22a2

. (8)

Proof. According to (4) the vector y = (x21, x

22, x

23, x1x2, x1x3, x2x3)

T satisfiesthe following equation in R6:

dy

dt= Ay + ξ(t)By, (9)

5

where

A =

0 0 0 2 0 00 −2γ 0 −2 0 −2k10 0 −2k2 0 0 2−1 1 0 −γ −k1 00 0 0 1 −k2 10 1 −k1 0 −1 −γ − k2

,

B =

0 0 0 0 0 00 −2σ 0 0 0 00 0 0 0 0 00 0 0 −σ 0 00 0 0 0 0 00 0 0 0 0 −σ

.

(10)

Then for the mean value E[y(t)] we have the coupled system [23, 24]

dE[y(t)]

dt= Ay +By1,

dy1(t)

dt= −αy1 + Ay1 +BE[y], (11)

where y1(t) = E[ξ(t)y(t)]. The mean square asymptotic stability of the triv-ial solution of system (4) with η ≡ 0 is equivalent to asymptotic stability ofthe trivial solution of system (9). This system is a set of 12 linear differentialequations of first order with the constant coefficients. It is well known thatasymptotic stability of its trivial solution can be investigated by inspectingthe signs of the real parts of the eigenvalues of the coefficient matrix. Usingwell–known Routh-Hurwitz criterion and symbolic computation packages onecan obtain the following necessary and sufficient condition of the stability:

g(σ) = a2σ4 − a1σ

2 + a0 > 0. (12)

There is a more simple approach to get this condition which was proposedin Ref. [25]. Note that 8k2g(σ) is a determinant of the coefficient matrix forsystem (9). Solving the inequality (12) one arrives to condition (8).

Let’s consider now system (5) with an electromechanical coupling. Themain quantity of interest in the energy harvesting is the electrical net meanpower [16–19]

P (t) =E[V 2(t)]

R.

6

Its non-dimensional counterpart is

P (t) =E[x2

3]

RC2p

= k1k2E[x3(t)]2

but in many application the long-term behavior of this value (stationarymean power)

Pst = k1k2 limt→∞

E[x3(t)]2 (13)

is very important.Theorem. If condition (7) is fulfilled then

Pst = k1k2Dg1(σ)

g(σ), (14)

where

g1(σ) = c1σ2 + c0, c1 = 2(α+ 2k2)[α

3 + 2k2 + α2(γ + k2) + α(1 + k1 + γk2)],

c0 = [(α + γ)(1 + α2 + αγ + k1) + (α + γ)(2α + γ)k2 + k1k2 + (α + γ)k22]

×[8k2 + 4α(1 + k1) + α(α + 2γ)(α + 2k2)].

Proof. Because the telegraphic process ξ(t) and the Wiener process w(t) areindependent the second moments can be computed in two step

E[xi(t)xj(t)] = Eξ[Ew[xi(t)xj(t)]], i, j = 1, 2, 3,

where Eξ and Ew are averaging over measure corresponding to the telegraphicprocess and the Wiener process respectively.Let introduce the vector v(t)=

(Ew[x1]2, Ew[x2]

2, Ew[x3]2, Ew[x1x2], Ew[x1x3], Ew[x2x3])

T ,

where x1 = x1(t), x2 = x2(t), and x3 = x3(t) satisfy set of equations (5).This vector satisfies the following equation in R6 [26]:

v = Av + ξ(t)Bv + d, (15)

where d = (0, 2D, 0, 0, 0, 0)T . Using again results from Refs. [23, 24] we have

dEξ[v(t)]

dt= AEξ[v(t)] +Bv1(t) + d,

7

dv1(t)

dt= −αv1(t) + Av1(t) +BEξ[v(t)], (16)

where v1(t) = Eξ[ξ(t)v(t)]. Therefore we get a system of 12 linear differ-ential equations with constant coefficients. Under condition (7) there existstationary moments that provide solution to system (5) and they satisfy thesystem of 12 linear algebraic equations associated with system (14). Apply-ing symbolic computation packages to this system one can obtain expression(12) for Pst.

From expression (12) follows an important conclusion that the value Ptends to infinity if σ → σ∗−, i.e. if σ tends to the stability boundary. Inthe left part of Fig. 2 the mean square stability chart for the system (4)with η ≡ 0 is presented but the right part is the plot of Pst. One can observesubstantial enhancement of the mean power near the stability boundary. Theapproach can be extended to some others models of the ambient excitationη(t) but system (14) is much more complicated for them.

It is interesting to note that the power of the parametric noise does nothave to be high. Indeed, if a ratio of the electrical efficiencies is considered,Θp/ΘL, where the efficiency in the parametrically excited system is Θp andefficiency of the linear system ΘL = Θp(σ = 0) and

Θ =Power(out)

Power(in),

then:Θp

ΘL

=Pst(σ)

Wη +Wξ

Wξ

Pst(0)=

Pst(σ)

Pst(0)

1

1 +Wη/Wξ

(17)

where Wξ and Wη are the power of the external and parametric noise respec-tively. In the case of unfeasible in reality a white noise, the second fractionis always equal to unity, therefore the efficiency of the parametric system isproportional to the power gained by it, which is much higher than that ofthe linear system. However, since a white noise is an approximation of a realphysical process, the latter has a finite power, and thus the ratio Wη/Wξ > 0making the second fraction less than unity. Hence, keeping the parametricsystem stable but close to the instability boundary, it is preferable to keepWη << Wξ to maximize the efficiency of the parametric system.

4. Conclusion

The paper proposes a new approach for substantial enhancement of energyharvesting using the electromechanical oscillator with a specially selected

8

random variations of the damping parameter. The method depends weaklyon the ambient excitation because the mean square stability boundary playscrucial role here. It may be considered as an advantage as compared withthe frequently used the linear resonator approach. However the randomvariations of the damping parameter is an active means which require somebut not much input power compare to the benefits of harvested energy. It ispossible to show that the proposed methodology can be extended to a widerclass of exponentially correlated processes.

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bHΤL

m

x+ Z-

ΘV

k

Z-

HΤL

R V

-

+

Θx

Cp

Figure 1: A simplified representation of the vibration - based piezoelectric energy harvester

0.5 1.0 1.5 2.0 2.5 3.0Α

0.2

0.4

0.6

0.8

1.0Σ

Unstable

Stable

0.0 0.1 0.2 0.3 0.4 Σ

5

10

15

20

25

30

Pst

Figure 2: The mean-square stability chart (left) and the stationary mean power (right)for the system (5) with parameter values γ = 0.1, k1 = 0.5, k2 = 0.3. In the right partα = D = 1.

12