Piezoelectric Energy Harvesting Under Uncertainty S Adhikari 1 M I - - PowerPoint PPT Presentation

Piezoelectric Energy Harvesting Under Uncertainty

S Adhikari1 M I Friswell1 D J Inman2

1School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP

, UK

2CIMSS, 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

Introduction Brief review of piezoelectric energy harvesting The role of uncertainty

2

Brief Overview of Stationary Random Vibration

3

Single Degree of Freedom Electromechanical Model Circuit without an inductor Circuit 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 powering low energy electronic devices has formed the focus of much recent research [1–6]. Of the published results that focus on the piezoelectric effect as the transduction method, almost all have focused on harvesting using cantilever beams and on single frequency ambient energy, i.e., resonance based energy harvesting. Soliman et al. [7] considered energy harvesting under wide band

• excitation. Liu et al. [8] proposed acoustic energy harvesting using

an electro-mechanical resonator. Shu et al. [9–11] conducted detailed analysis of the power output for piezoelectric energy harvesting systems. Several authors [12–15, 15] have proposed methods to optimize the parameters of the system to maximize the 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 input excitation may not be always known exactly. There may be uncertainties associated with the numerical values considered for various parameters of the harvester. This might arise, for example, due to the difference between the true values and the assumed values. If there are several nominally identical energy harvesters to be manufactured, there may be genuine parametric variability within the ensemble. Any deviations from the assumed excitation may result an

• ptimally 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: L{u(t)} = f(t) (1) Uncertainty in the input excitations For this case in general f(t) is a random function of time. Such functions are called random processes. In this work we consider stationary Gaussian random processes, characterised by the standard deviation σ and two-point autocorrelation function R(t1, t2). Uncertainty in the system The operator L{•} is in general a function of parameters θ1, θ2, · · · , θn ∈ R. The uncertainty in the system can be characterised by the joint probability 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 been discussed by Lin [16], Nigam [17], Bolotin [18], Roberts and Spanos [19] and Newland [20] within the scope of random vibration theory. When xb(t) is a weakly stationary random process, its autocorrelation function depends only on the difference in the time instants: E [xb(τ1)xb(τ2)] = Rxbxb(τ1 − τ2). (2) This autocorrelation function can be expressed as the inverse Fourier 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 be shown that [16, 17] the spectral density of V is related to the spectral density 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 two cases considered. We assume that the base acceleration ¨ xb(t) is Gaussian white noise so that its spectral density is constant with respect 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 calculation

• f 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−2ω2n−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 ··· ··· −an−1 an−3 −an−5 ··· ··· an −an−2 an−4 ··· ··· ··· ··· ··· ··· −a2 a0

    (11) and Nn =    

an−1 −an−3 an−5 −an−7 −an an−2 −an−4 an−6 ··· ··· −an−1 an−3 −an−5 ··· ··· an −an−2 an−4 ··· ··· ··· ··· ··· ··· −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 ProofMass x xb

• +

v Rl Base Piezo- ceramic ProofMass x xb

• +

v Rl L

Schematic diagrams of piezoelectric energy harvesters with two different harvesting 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 electromechanical behavior by the linear ordinary differential equations m¨ x(t) + c ˙ x(t) + kx(t) − θv(t) = −m ¨ xb(t) (13) θ ˙ x(t) + Cp ˙ v(t) + 1 Rl v(t) = 0 (14) Transforming both the equations into the frequency domain and dividing the first equation by m and the second equation by Cp we

• btain
• −ω2 + 2iωζωn + ω2

n

• X(ω) − θ

mV(ω) = ω2Xb(ω) (15) iω θ Cp X(ω) +

• iω +

1 CpRl

• 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 =

• k

m and ζ = c 2mωn . (17) Dividing the preceding equations by ωn and writing in matrix form one has

• 1 − Ω2

+ 2iΩζ − θ

k

iΩ αθ

Cp

(iΩα + 1) X V

• =

Ω2Xb

• ,

(18) where the dimensionless frequency and dimensionless time constant are defined as Ω = ω ωn and α = ωnCpRl. (19) α is the time constant of the first order electrical system, non-dimensionalized using the natural frequency of the mechanical system.

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 the frequency domain can be obtained as X

V

• = 1

∆1

• (iΩα+1)

θ k

−iΩ αθ

Cp (1−Ω2)+2iΩζ

Ω2Xb

• =
• (iΩα+1)Ω2Xb/∆1

−iΩ3 αθ

Cp 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 Theorem The average harvested power due to the white-noise base acceleration with a circuit with an inductor is given by E

• P
• = E
• |V|2

(Rlω4Φxbxb)

• =

π m α κ2

(2 ζ α2+α)κ2+4 ζ2α+(2 α2+2)ζ .

1

From Equation (20) we obtain the voltage in the frequency domain as V = −iΩ3 αθ

Cp

∆1(iΩ) Xb. (23)

2

Following DuToit and Wardle [21] we are interested in the mean of the normalized harvested power when the base acceleration is Gaussian white noise, that is |V|2/(Rlω4Φxbxb).

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 be

• constant. After some algebra, from Equation (23), the normalized

power is

• P =

|V|2 (Rlω4Φxbxb) = kακ2 ω3

n

Ω2 ∆1(iΩ)∆∗

1(iΩ).

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

• P
• = E
• |V|2

(Rlω4Φxbxb)

• = kακ2

ω3

n

∞

−∞

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 average harvested 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     1 −α α + κ2α + 2 ζ −2 ζ α − 1 1     det     2 ζ α + 1 −1 −α α + κ2α + 2 ζ −2 ζ α − 1 1     (29)

Combining this with Equation (26) we obtain the average harvested

Adhikari, 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

0.1 0.2 1 2 3 4 1 2 3 4 5

α ζ Normalized mean power

The normalized mean power of a harvester with an inductor as a function of α and β, with ζ = 0.1 and κ = 0.6. Maximizing the average power with respect to α gives the condition α2 1 + κ2 = 1 or in terms

• f physical quantities R2

l 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 θ¨ x(t) + Cp¨ v(t) + 1 Rl ˙ v(t) + 1 Lv(t) = 0 (30) where L is the inductance of the circuit. Transforming equation (30) into the frequency domain and dividing by Cpω2

n one has

− Ω2 θ Cp X +

• −Ω2 + iΩ 1

α + 1 β

• V = 0

(31) where the second dimensionless constant is defined as β = ω2

nLCp,

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

− θ

k

−Ω2 αβθ

Cp

α(1−βΩ2)+iΩβ

X

V

• =
• Ω2Xb
• .

(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 the frequency domain can be obtained as X

V

• = 1

∆2

• α(1−βΩ2)+iΩβ

θ k

Ω2 αβθ

Cp

(1−Ω2)+2iΩζ

Ω2Xb

• =

(α(1−βΩ2)+iΩβ)Ω2Xb/∆2

Ω4 αβθ

Cp Xb/∆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 Theorem The average harvested power due to the white-noise base acceleration 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

1 2 3 4 1 2 3 4 0.5 1 1.5

β α Normalized mean power

The normalized mean power of a harvester with an inductor as a function 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 1 2 3 4 0.5 1 1.5

β Normalized mean power

The normalized mean power of a harvester with an inductor as a function of β for α = 0.6, ζ = 0.1 and κ = 0.6. The * corresponds to the 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 1 2 3 4 0.5 1 1.5

α Normalized mean power

The normalized mean power of a harvester with an inductor as a function of α for β = 1, ζ = 0.1 and κ = 0.6. The * corresponds to the

• ptimal 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 are expected to operate under a wide range of ambient environments. This talk considers energy harvesting of such systems under broadband random excitations.

2

Specifically, analytical expressions of the normalized mean harvested power due to stationary Gaussian white noise base excitation has been derived.

3

Two cases, namely the harvesting circuit with and without an inductor, have been considered.

4

It was observed that in order to maximise the mean of the harvested power (a) the mechanical damping in the harvester should be minimized, and (b) the electromechanical coupling should be as large as possible.

Adhikari, Friswell, Inman (Swansea & VT) Energy Harvesting Under Uncertainty 17 December 2009 24 / 26

Summary & Future Directions

References

[1] Sodano, H., Inman, D., and Park, G., 2004, “A review of power harvesting from vibration using piezoelectric materials,” The Shock 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),” Smart Materials & Structures, 16(3), pp. R1–R21. [5] Lefeuvre, E., Badel, A., Benayad, A., Lebrun, L., Richard, C., and Guyomar, D., 2005, “A comparison between several approaches of piezoelectric energy harvesting,” Journal de Physique IV, 128, pp. 177–186, Meeting on Electro-Active Materials 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-powered piezoelectric 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 energy harvester,” 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 harvesting using 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 of Micromechanics 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 of Intelligent 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 energy harvesters,” 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 and Structures, 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 of Sound 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