piezoelectric energy harvesting under...

Piezoelectric Energy Harvesting UnderUncertainty

S Adhikari1 M I Friswell1 D J Inman2

1School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK2CIMSS, Department of Mechanical Engineering, Virginia Polytechnic Institute and State

University, Blacksburg, VA, USA

Bristol Energy Harvesting Workshop

Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 1 / 26

Outline

1 IntroductionBrief review of piezoelectric energy harvestingThe role of uncertainty

2 Brief Overview of Stationary Random Vibration

3 Single Degree of Freedom Electromechanical ModelCircuit without an inductorCircuit with an inductor

4 Summary & Future Directions


Introduction Brief review of piezoelectric energy harvesting

Brief review of piezoelectric energy harvesting

The harvesting of ambient vibration energy for use in poweringlow energy electronic devices has formed the focus of muchrecent research [1–6].

Of the published results that focus on the piezoelectric effect asthe transduction method, almost all have focused on harvestingusing cantilever beams and on single frequency ambient energy,i.e., resonance based energy harvesting.

Soliman et al. [7] considered energy harvesting under wide bandexcitation. Liu et al. [8] proposed acoustic energy harvesting usingan electro-mechanical resonator. Shu et al. [9–11] conducteddetailed analysis of the power output for piezoelectric energyharvesting systems. Several authors [12–15, 15] have proposedmethods to optimize the parameters of the system to maximizethe harvested energy.


Introduction The role of uncertainty

Why uncertainty is important for energy harvesting?

In the context of energy harvesting of ambient vibration, the inputexcitation may not be always known exactly.

There may be uncertainties associated with the numerical valuesconsidered for various parameters of the harvester. This mightarise, for example, due to the difference between the true valuesand the assumed values.

If there are several nominally identical energy harvesters to bemanufactured, there may be genuine parametric variability withinthe ensemble.

Any deviations from the assumed excitation may result anoptimally designed harvester to become sub-optimal.


Introduction The role of uncertainty

Types of uncertainty

Suppose the set of coupled equations for energy harvesting:

Lu(t) = f(t) (1)

Uncertainty in the input excitations

For this case in general f(t) is a random function of time. Suchfunctions are called random processes.

In this work we consider stationary Gaussian random processes,characterised by the standard deviation σ and two-pointautocorrelation function R(t1, t2).

Uncertainty in the system

The operator L• is in general a function of parametersθ1, θ2, · · · , θn ∈ R.

The uncertainty in the system can be characterised by the jointprobability density function pΘ1,Θ2,··· ,Θn (θ1, θ2, · · · , θn).


Brief Overview of Stationary Random Vibration

Stationary random vibration

Mechanical systems driven by this type of excitation have beendiscussed by Lin [16], Nigam [17], Bolotin [18], Roberts andSpanos [19] and Newland [20] within the scope of randomvibration theory.

When xb(t) is a weakly stationary random process, itsautocorrelation function depends only on the difference in the timeinstants:

E [xb(τ1)xb(τ2)] = Rxbxb(τ1 − τ2). (2)

This autocorrelation function can be expressed as the inverseFourier transform of the spectral density Φxbxb(ω) as

Rxbxb(τ1 − τ2) =

∫ ∞

−∞Φxbxb(ω)exp[iω(τ1 − τ2)]dω. (3)




We are interested in the average harvested power given by

E [P(t)] = E

[v2(t)

Rl

]=

E[v2(t)

]

Rl. (4)

For a damped linear system of the form V (ω) = H(ω)Xb(ω), it can beshown that [16, 17] the spectral density of V is related to the spectraldensity of Xb by

ΦVV (ω) = |H(ω)|2Φxbxb(ω). (5)

Thus, for large t , we obtain

E

[v2(t)

]= Rvv (0) =

∫ ∞

−∞|H(ω)|2Φxbxb(ω) dω. (6)

This expression will be used to obtain the average power for the twocases considered. We assume that the base acceleration xb(t) isGaussian white noise so that its spectral density is constant withrespect to frequency.




The calculation of the preceding expressions requires the calculationof integrals of the following form

In =

∫ ∞

−∞

Ξn(ω) dω

Λn(ω)Λ∗n(ω)

(7)

where the polynomials have the form

Ξn(ω) = bn−1ω2n−2 + bn−2ω

2n−4 + · · ·+ b0 (8)

Λn(ω) = an(iω)n + an−1(iω)

n−1 + · · ·+ a0 (9)

Following Roberts and Spanos [19] this integral can be evaluated as

In =π

an

det [Dn]

det [Nn]. (10)




In the preceding expression the m × m matrices are

Dn =

bn−1 bn−2 ··· b0−an an−2 −an−4 an−6 ··· 0 ···

0 −an−1 an−3 −an−5 ··· 0 ···0 an −an−2 an−4 ··· 0 ···0 ··· ··· 0 ···0 0 ··· −a2 a0

(11)

and

Nn =

an−1 −an−3 an−5 −an−7−an an−2 −an−4 an−6 ··· 0 ···

0 −an−1 an−3 −an−5 ··· 0 ···0 an −an−2 an−4 ··· 0 ···0 ··· ··· 0 ···0 0 ··· −a2 a0

. (12)


Single Degree of Freedom Electromechanical Model

SDOF electromechanical models

Base

Piezo-

ceramic

Proof Mass

x

xb

-

+

vRl

Base

Piezo-

ceramic

Proof Mass

x

xb

-

+

vRlL

Schematicdiagrams of piezoelectric energy harvesters with two differentharvesting circuits. (a) Harvesting circuit without an inductor, (b)Harvesting circuit with an inductor.


Single Degree of Freedom Electromechanical Model Circuit without an inductor

Circuit without an inductor

DuToit and Wardle [21] expressed the coupled electromechanicalbehavior by the linear ordinary differential equations

mx(t) + cx(t) + kx(t)− θv(t) = −mxb(t) (13)

θx(t) + Cpv(t) +1Rl

v(t) = 0 (14)

Transforming both the equations into the frequency domain anddividing the first equation by m and the second equation by Cp weobtain

(−ω2 + 2iωζωn + ω2

n

)X (ω)−

θ

mV (ω) = ω2Xb(ω) (15)

iωθ

CpX (ω) +

(iω +

1CpRl

)V (ω) = 0 (16)




The natural frequency of the harvester, ωn, and the damping factor, ζ,are defined as

ωn =

√km

and ζ =c

2mωn. (17)

Dividing the preceding equations by ωn and writing in matrix form onehas [(

1 − Ω2)+ 2iΩζ − θ

kiΩαθ

Cp(iΩα+ 1)

]XV

=

Ω2Xb

0

, (18)

where the dimensionless frequency and dimensionless time constantare defined as

Ω =ω

ωnand α = ωnCpRl . (19)

α is the time constant of the first order electrical system,non-dimensionalized using the natural frequency of the mechanicalsystem.




Inverting the coefficient matrix, the displacement and voltage in thefrequency domain can be obtained as

XV

=

1∆1

[(iΩα+1) θ

k

−iΩαθCp

(1−Ω2)+2iΩζ

]Ω2Xb

0

=

(iΩα+1)Ω2Xb/∆1

−iΩ3 αθCp

Xb/∆1

, (20)

where the determinant of the coefficient matrix is

∆1(iΩ) = (iΩ)3α+ (2 ζ α+ 1) (iΩ)2 +(α+ κ2α+ 2 ζ

)(iΩ) + 1 (21)

and the non-dimensional electromechanical coupling coefficient is

κ2 =θ2

kCp. (22)




TheoremThe average harvested power due to the white-noise baseacceleration with a circuit with an inductor is given by

E

[P]= E

[|V |2

(Rlω4Φxbxb )

]= π mακ2

(2 ζ α2+α)κ2+4 ζ2α+(2α2+2)ζ.

1 From Equation (20) we obtain the voltage in the frequency domainas

V =−iΩ3 αθ

Cp

∆1(iΩ)Xb. (23)

2 Following DuToit and Wardle [21] we are interested in the mean ofthe normalized harvested power when the base acceleration isGaussian white noise, that is |V |2/(Rlω

4Φxbxb).




The spectral density of the acceleration ω4Φxbxb) and is assumed to beconstant. After some algebra, from Equation (23), the normalizedpower is

P =|V |2

(Rlω4Φxbxb)=

kακ2

ω3n

Ω2

∆1(iΩ)∆∗1(iΩ)

. (24)

Using Equation (6), the average normalized power can be obtained as

E

[P]= E

[|V |2

(Rlω4Φxbxb)

]=

kακ2

ω3n

∫ ∞

−∞

Ω2

∆1(iΩ)∆∗1(iΩ)

dω (25)

From Equation (21) observe that ∆1(iΩ) is third order polynomial in(iΩ). Noting that dω = ωndΩ and from Equation (21), the averageharvested power can be obtained from Equation (25) as

E

[P]= E

[|V |2

(Rlω4Φxbxb)

]= mακ2I(1) (26)




I(1) =∫ ∞

−∞

Ω2

∆1(iΩ)∆∗1(iΩ)

dΩ. (27)

Comparing I(1) with the general integral in equation (7) we have

n = 3,b2 = 0,b1 = 1,b0 = 0

and a3 = α,a2 = (2 ζ α+ 1) ,a1 =(α+ κ2α+ 2 ζ

),a0 = 1

(28)

Now using Equation (10), the integral can be evaluated as

I(1) =π

α

det

0 1 0

−α α+ κ2α+ 2 ζ 0

0 −2 ζ α− 1 1

det

2 ζ α+ 1 −1 0

−α α+ κ2α+ 2 ζ 0

0 −2 ζ α− 1 1

(29)

Combining this with Equation (26) we obtain the average harvestedAdhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 16 / 26


Normalised mean power: numerical illustration

0

0.1

0.2 01

23

40

1

2

3

4

5

αζ

Norm

alize

d mea

n pow

er

The normalized mean power of a harvester with an inductor as afunction of α and β, with ζ = 0.1 and κ = 0.6. Maximizing the averagepower with respect to α gives the condition α2

(1 + κ2

)= 1 or in terms

of physical quantities R2l Cp

(kCp + θ2

)= m.


Single Degree of Freedom Electromechanical Model Circuit with an inductor

Circuit with an inductor

Following Renno et al. [15], the electrical equation becomes

θx(t) + Cpv(t) +1Rl

v(t) +1L

v(t) = 0 (30)

where L is the inductance of the circuit. Transforming equation (30)into the frequency domain and dividing by Cpω

2n one has

− Ω2 θ

CpX +

(−Ω2 + iΩ

1α+

1β

)V = 0 (31)

where the second dimensionless constant is defined as

β = ω2nLCp, (32)

Two equations can be written in a matrix form as[(1−Ω2)+2iΩζ − θ

k

−Ω2 αβθ

Cpα(1−βΩ2)+iΩβ

]XV

=

Ω2Xb

0

. (33)




Inverting the coefficient matrix, the displacement and voltage in thefrequency domain can be obtained as

XV

=

1∆2

[α(1−βΩ2)+iΩβ θ

k

Ω2 αβθ

Cp(1−Ω2)+2iΩζ

]Ω2Xb

0

=

(α(1−βΩ2)+iΩβ)Ω2Xb/∆2

Ω4 αβθ

CpXb/∆2

(34)

where the determinant of the coefficient matrix is

∆2(iΩ) = (iΩ)4β α+ (2 ζ β α+ β) (iΩ)3

+(β α+ α+ 2 ζ β + κ2β α

)(iΩ)2 + (β + 2 ζ α) (iΩ) + α. (35)




TheoremThe average harvested power due to the white-noise baseacceleration with a circuit with an inductor is given by

E

[P]= mαβκ2π(β+2αζ)

β(β+2αζ)(1+2αζ)(ακ2+2ζ)+2α2ζ(β−1)2 .

1 Proof is very similar to the previous case.



Normalised mean power: numerical illustration

01

23

4

01

23

40

0.5

1

1.5

βα

Norm

alize

d mea

n pow

er

The normalized mean power of a harvester with an inductor as afunction of α and β, with ζ = 0.1 and κ = 0.6.



Parameter selection

0 1 2 3 40

0.5

1

1.5

β

Nor

mal

ized

mea

n po

wer

The normalized mean power of a harvester with an inductor as afunction of β for α = 0.6, ζ = 0.1 and κ = 0.6. The * corresponds tothe optimal value of β(= 1) for the maximum mean harvested power.



Parameter selection

0 1 2 3 40

0.5

1

1.5

α

Nor

mal

ized

mea

n po

wer

The normalized mean power of a harvester with an inductor as afunction of α for β = 1, ζ = 0.1 and κ = 0.6. The * corresponds to theoptimal value of α(= 1.667) for the maximum mean harvested power.


Summary & Future Directions

Summary of results

1 Vibration energy based piezoelectric energy harvesters areexpected to operate under a wide range of ambient environments.This talk considers energy harvesting of such systems underbroadband random excitations.

2 Specifically, analytical expressions of the normalized meanharvested power due to stationary Gaussian white noise baseexcitation has been derived.

3 Two cases, namely the harvesting circuit with and without aninductor, have been considered.

4 It was observed that in order to maximise the mean of theharvested power (a) the mechanical damping in the harvestershould be minimized, and (b) the electromechanical couplingshould be as large as possible.


Summary & Future Directions

References

[1] Sodano, H., Inman, D., and Park, G., 2004, “A review of power harvesting from vibration using piezoelectric materials,” TheShock and Vibration Digest, 36, pp. 197–205.

[2] Beeby, S. P., Tudor, M. J., and White, N. M., 2006, “Energy harvesting vibration sources for microsystems applications,”Measurement Science & Technology, 17(12), pp. R175–R195.

[3] Priya, S., 2007, “Advances in energy harvesting using low profile piezoelectric transducers,” Journal of Electroceramics,19(1), pp. 165–182.

[4] Anton, S. R., and Sodano, H. A., 2007, “A review of power harvesting using piezoelectric materials (2003-2006),” SmartMaterials & Structures, 16(3), pp. R1–R21.

[5] Lefeuvre, E., Badel, A., Benayad, A., Lebrun, L., Richard, C., and Guyomar, D., 2005, “A comparison between severalapproaches of piezoelectric energy harvesting,” Journal de Physique IV, 128, pp. 177–186, Meeting on Electro-ActiveMaterials and Sustainable Growth, Abbey les Vaux de Cernay, FRANCE, May 23-25, 2005.

[6] Lefeuvre, E., Badel, A., Richard, C., Petit, L., and Guyomar, D., 2006, “A comparison between several vibration-poweredpiezoelectric generators for standalone systems,” Sensors and Actuators A-Physical, 126(2), pp. 405–416.

[7] Soliman, M. S. M., Abdel-Rahman, E. M., El-Saadany, E. F., and Mansour, R. R., 2008, “A wideband vibration-based energyharvester,” Journal of Micromechanics and Microengineering, 18(11).

[8] Liu, F., Phipps, A., Horowitz, S., Ngo, K., Cattafesta, L., Nishida, T., and Sheplak, M., 2008, “Acoustic energy harvestingusing an electromechanical Helmholtz resonator,” Journal of the Acoustical Society of America, 123(4), pp. 1983–1990.

[9] Shu, Y. C., and Lien, I. C., 2006, “Efficiency of energy conversion for a piezoelectric power harvesting system,” Journal ofMicromechanics and Microengineering, 16(11), pp. 2429–2438.

[10] Shu, Y. C., and Lien, I. C., 2006, “Analysis of power output for piezoelectric energy harvesting systems,” Smart Materials &Structures, 15(6), pp. 1499–1512.

[11] Shu, Y. C., Lien, I. C., and Wu, W. J., 2007, “An improved analysis of the SSHI interface in piezoelectric energy harvesting,”Smart Materials & Structures, 16(6), pp. 2253–2264.

[12] Ng, T., and Liao, W., 2005, “Sensitivity analysis and energy harvesting for a self-powered piezoelectric sensor,” Journal ofIntelligent Material Systems and Structures, 16(10), pp. 785–797.

[13] duToit, N., Wardle, B., and Kim, S., 2005, “Design considerations for MEMS-scale piezoelectric mechanical vibration energyharvesters,” Integrated Ferroelectrics, 71, pp. 121–160, 13th International Materials Research Congress (IMRC), Cancun,Mexico, August 22-26, 2004.

[14] Roundy, S., 2005, “On the effectiveness of vibration-based energy harvesting,” Journal of Intelligent Material Systems andStructures, 16(10), pp. 809–823.

[15] Renno, J. M., Daqaq, M. F., and Inman, D. J., 2009, “On the optimal energy harvesting from a vibration source,” Journal ofSound and Vibration, 320(1-2), pp. 386–405.

[16] Lin, Y. K., 1967, Probabilistic Thoery of Strcutural Dynamics, McGraw-Hill Inc, Ny, USA.[17] Nigam, N. C., 1983, Introduction to Random Vibration, The MIT Press, Cambridge, Massachusetts.[18] Bolotin, V. V., 1984, Random vibration of elastic systems, Martinus and Nijhoff Publishers, The Hague.[19] Roberts, J. B., and Spanos, P. D., 1990, Random Vibration and Statistical Linearization, John Wiley and Sons Ltd,

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