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
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 2 / 26
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.
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 3 / 26
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.
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 4 / 26
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).
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 5 / 26
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)
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 6 / 26
Brief Overview of Stationary Random Vibration
Stationary random vibration
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.
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 7 / 26
Brief Overview of Stationary Random Vibration
Stationary random vibration
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)
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 8 / 26
Brief Overview of Stationary Random Vibration
Stationary random vibration
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)
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 9 / 26
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.
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 10 / 26
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)
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 11 / 26
Single Degree of Freedom Electromechanical Model Circuit without an inductor
Circuit without an inductor
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.
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 12 / 26
Single Degree of Freedom Electromechanical Model Circuit without an inductor
Circuit without an inductor
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)
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 13 / 26
Single Degree of Freedom Electromechanical Model Circuit without an inductor
Circuit without an inductor
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).
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 14 / 26
Single Degree of Freedom Electromechanical Model Circuit without an inductor
Circuit without an inductor
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)
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 15 / 26
Single Degree of Freedom Electromechanical Model Circuit without an inductor
Circuit without an inductor
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
Single Degree of Freedom Electromechanical Model Circuit without an inductor
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.
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 17 / 26
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)
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 18 / 26
Single Degree of Freedom Electromechanical Model Circuit with an inductor
Circuit with an inductor
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)
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 19 / 26
Single Degree of Freedom Electromechanical Model Circuit with an inductor
Circuit with an inductor
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.
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 20 / 26
Single Degree of Freedom Electromechanical Model Circuit with an inductor
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.
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 21 / 26
Single Degree of Freedom Electromechanical Model Circuit with an inductor
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.
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 22 / 26
Single Degree of Freedom Electromechanical Model Circuit with an inductor
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.
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 23 / 26
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.
Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 24 / 26
Summary & Future Directions
References
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[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.
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[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.
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[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,
Chichester, England.Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 25 / 26