Energy Harvesting in Piezoelastic Systems

Driven by Random Excitations

M. Borowiec, G. Litak∗

Faculty of Mechanical Engineering, Lublin University of Technology,

PL-20-618 Lublin, Poland

M.I. Friswell, S.F. Ali, S. Adhikari, A.W. Lees, O. BilgenCollege of Engineering, Swansea University, Singleton Park,

Swansea SA2 8PP, UK

Abstract

The inverted elastic beam is proposed as an energy harvester. The beam has a

tip mass and piezoelectric layers which transduce the bending strains induced by

the stochastic horizontal displacement into electrical charge. The efficiency of this

nonlinear device is analyzed, focusing on the region of stochastic resonance where

the beam motion has a large amplitude. Increasing the noise level allows the motion

of the beam system to escape from single well oscillations and thus generate more

power.

Number of figures: 18Total number of pages in the manuscript: 11

Manuscript accepted to: International Journal of Structural Stability andDynamics (IJSSD) August 1, 2012

∗Corresponding author: Grzegorz Litak, Email:g.litak@pollub.pl, Tel: +48 815384573, Fax: +48

815384233

1

1 IntroductionRecently, the demands on portable devices, self-powered wireless sensors and simi-

lar applications that currently require rechargeable batteries, have increased considerably[1, 2, 3, 4]. Vibration energy is one of the candidate technologies, in competition withother possibilities such as photo-voltaic or thermal gradient approaches [6, 7]. The effec-tiveness of devices based on harvesting vibration energy and their availability are con-tinuously growing [5]. Ambient mechanical vibrations can be converted to the electricalenergy using electromagnetic, piezoelectric, and electrostatic methods [1, 5, 6]. Piezoelec-tric transducers can be easily integrated into elastic multilayer cantilever beams; Priya[8] reviewed potential piezoelectric materials for energy harvesting and discusses variousprototypes.

The main advantage of piezoelectric devices, compared to electrostatic and electro-magnetic devices, is their high power output for a given device volume and excitation[8, 7]. Although not suitable for generating high power levels, they are very suitable forapplications to small electronic devices. The main issue with this form of harvester isthe source of ambient energy. The classic approach is to tune the mechanical part of theharvester to resonate; this approach is good for narrow band excitations but performspoorly in off-resonance conditions or with broadband ambient excitation [1, 4]. Unfortu-nately, the small levels of energy available off-resonance is often insufficient and a tuningprocess has to be employed. Ambient excitation forces, such as wind and road trafficmotion, often provide changeable conditions characterized by fluctuating frequency andamplitude, or by broadband excitation. Thus, the ideal device is a broadband harvesterwhich is not critically sensitive to the specific frequency or amplitude of excitation [6, 7].One approach is a nonlinear mechanical resonator with a dual well potential, obtainedusing a magnetic field or special prestressed conditions [9, 10, 11, 12].

Recently, Friswell at al. [13] studied a piezoelastic system consisting of a cantileverbeam with a tip mass that was mounted vertically and harmonically excited in the trans-verse direction at its base. For a relatively large tip mass the vertical position was unstableand the beam buckled, giving a double well potential due to gravitational loading. Thedeterministic model was simulated and validated using an experimental device with threedifferent tip masses, representing three interesting cases: a linear system; a low naturalfrequency, non-buckled beam; and a buckled beam. The most practical configuration ap-peared to be the pre-buckled case, where the proposed system has a low natural frequency,a high level of harvested power, and an increased bandwidth over a linear harvester.

In the present paper we follow the above mentioned model (Friswell at al. [13]) andcomplete the analysis by simulating an inverted beam excited by random noise. Similarapproaches to a broadband piezoelastic device using random noise excitation have beendiscussed by [11, 14, 15, 16, 17, 18, 19, 20]. Note also that the nonlinear system consideredin this paper bends due to the gravitational field (Fig. 1), in contrast to the magneticsystems discussed in previous papers [11, 14, 15, 16, 17, 18, 19, 20]. More recent analysison nonlinear beam system vibrations can be found in [21, 22, 23].

2

x

v

u

g

mt

y

L

Figure 1: The model of the inverted beam with a tip mass mt with its horizontal andvertical displacements v and u. L is the length of beam, while x is the axis oriented alongthe beam. g represents the gravitational acceleration while, y denotes the kinematichorizontal excitation.

2 Equations of MotionFigure 1 shows a schematic of the system considered, which consists of an inverted

beam with one end fixed to the moving base and other loaded by a tip mass, mt. Theequation of motion has been derived using Lagrange’s method. An arbitrary point atthe beam denoted by mass element dm at distance x from the oscillating base and thepoint mass mt were considered, which undergo a flexible body deflection to a new positiondue to the base excitation. The mass element of the beam dm undergoes a horizontaldisplacement y + vdm and a vertical displacement (−udm) and a rotation of φ of the tipbeam point (Fig. 1). Note that in our calculations, the effect of rotary inertia of the beammass, φdm, was neglected.

The kinetic energy of the system is [13, 24, 25]

T =1

2ρA

∫

m

[(

vdm(x, t) + y(t))2

+ udm(x, t)2

]

dx

+1

2mt

[(

v(t) + y(t))2

+ u(t)2

]

+1

2I0φ

2 (1)

and the potential energy arising from the strain in the elastic beam, and gravity termsfrom the beam and tip mass, is

Π =1

2EI

∫ L

0

κ(x, t)2dx+ gρA

∫

m

udm(x, t)dx+mtgu(t). (2)

A single degree of freedom model of the deflection of the beam is considered, where

vdm(x, t) = v(t)ψ(x) (3)

3

Table 1: System parameters.symbol and value descriptionL = 0.2m length of the beammt = 0.038kg tip massρ = 7850kg/m3 density of the beam massA = 0.000004m2 cross section area of beamE = 210GPa Young’s modulusI = 2.15 × 10−14m4 geometrical moment of inertiaI0 = 1144.4 × 10−3 kgm2 mass moment of inertiay kinematic displacement of the baseCp = 38.9nF capacitance of the piezoelectric patchesR = 50kΩ load resistanceLp = 28mm active length of piezoelectric layersbb = 15mm beam widthhb = 2.5mm beam thicknesshp = 300µm piezoelectric layers thicknesse31 = −5.157 C/m2 piezoelectric constant

Table 2: Effective parameters in SI units (see Eqs. (4) - (7)); the corresponding definitionscan be found in the Appendix.

N1 N2 N3 N4 N5 N6 N7 N8 N9

0.0453521 0.072676 1.3801 6.1685 7.85398 380.504 5867.85 180979. 0.36685

for some known function ψ(x) that approximates the deformation in the beam. By ne-glecting terms higher than third order the horizontal deflection of the beam udm may bewritten in terms of the primary degree of freedom v. Thus the kinetic and potential ener-gies may be written in terms of v and Lagrange’s method applied to derive the equationof motion of the beam-mass system [13]

[

mt + ρAN1 + I0N2

5+

(

ρAN3 +mtN2

4+ I0N

4

5

)

v2

]

v

+[

ρAN3 +mtN2

4+ I0N

4

5

]

vv2

−

[

g(

mtN4 + ρAN9

)

− EI(

N6 + 2N7v2

)]

v = −

[

mt + ρAN2

]

y, (4)

where y is the acceleration of the base of the beam and is assumed to be stochastic andof the form

y = Γ(t), (5)

where Γ(t) denotes the stochastic factor. The other parameters used are defined in Tables1 and 2.

4

3 The Coupled Piezoelectric ModelUsing the derivation of the coupled electro-mechanical model provided by the authors

[13], the equation of motion combined with the electrical equation becomes[

mt + ρAN1 + I0N2

5+

(

ρAN3 +mtN2

4+ I0N

4

5

)

v2

]

v

+[

ρAN3 +mtN2

4+ I0N

4

5

]

vv2

−

[

g(

mtN4 + ρAN9

)

− EI(

N6 + 2N7v2

)]

v −DU = −

[

mt + ρAN2

]

y

CpU +U

R+Dv = 0, (6)

where U is the voltage across the load resistor connected to the piezoelectric patch, Cp

is the capacitance of the piezoelectric patches and R is the load resistance. Two piezo-electric patches are attached to the beam and connected in parallel. The constant for theelectromechanical coupling is given by

D = e31bp(hp + hb)

∫ Lp

0

ψ′′(x)dx = e31bp(hp + hb)ψ′(Lp), (7)

where hb is the beam thickness, hp is the piezoelectric layer thickness, bp is the piezoelectriclayer width, Lp denotes active length of piezoelectric layer, and e31 is the piezoelectricconstant.

4 Numerical Results and DiscussionBased on Eq. (6), numerical simulations of the beam system with attached piezoelec-

tric layers (Fig. 1) have been performed. We assumed a kinematic excitation, where ywas expressed as the random noise factor Γ(t) (Eq. (5)). In the simulations two differentkinds of noise were used; Gaussian noise with a normal distribution (ND), and noise witha uniform distribution (UD). The results of the simulations are presented in Figs. 2a-d.

Figure 2a shows the displacement signal to noise ratio σv/σF (in terms of the corre-sponding standard deviations) versus the input noise represented by σF for a tip mass ofmt=0.038kg. For both noise distributions (normal (ND) and uniform (UD)) one can ob-serve a clear maximum around σv/σF = 0.25. This maximum corresponds to a stochasticresonance [26], which gives the largest amplitude oscillation for a given excitation level,and reflects the transition in the system response from single potential well oscillations todouble well vibration characteristics with hopping between the two potential wells. Onecan note that for the normal noise excitation the transition happens for a smaller noiseintensity, σF , and this because of the long tails of the Gaussian white noise distribution.Such a tail can generate much larger accelerations than for the bounded uniform noisecase. Consequently, in the region of fairly small σv, the value of σv/σF is slightly larger forND than for UD. The variance of the voltage output, U (Fig. 2b), increases considerablyafter σF passes the stochastic resonance case.

Additionally, we plotted the number of hops between the potential wells (Fig. 2c).As mentioned earlier these are strictly connected to the appearance of the stochasticresonance. Note that the appearance of a larger number of hops in Fig. 2c coincidences

5

(a)0 0.1 0.2 0.3 0.4 0.5

0

0.1

0.2

0.3

0.4

0.5

σF

σ v /σF

NDUD

(b)0 0.1 0.2 0.3 0.4 0.5

0

2

4

6

8

10

12

x 10−4

σF

σ U2

NDUD

(c)0 0.1 0.2 0.3 0.4 0.5

0

10

20

30

40

50

σF

Num

ber

of h

ops

NDUD

(d)0 0.1 0.2 0.3 0.4 0.5

−0.1

−0.05

0

0.05

0.1

0.15

σF

Mea

n of

dis

plac

emen

t <v>

NDUD

Figure 2: The displacement signal to noise ratio σv/σF versus σF for tip mass mt=0.038kg(a), the variance of the generated voltage U for the corresponding tip mass mt (b), thenumber of hops (c) and the mean of displacement (d). Two different kinds of noise havebeen used: normal (ND) and uniform distributions (UD). The simulation time period wasfixed as t ∈ [0, 10800s], while the time step was δt = 0.01s. The initial time interval foreach simulation [0, 7200s] was neglected as a transient in the statistical estimation of thestandard deviation σF , the number of hops and mean value of displacement < v >. Theinitial conditions for each noise level were [v, v, U ]|t=0 = [0.1042m, 0, 0]. Each point on thefigures is an average of 10 different noise realizations.

with the maxima in Fig. 2a. The effect of this solution bifurcation from single to doublewell oscillations is also visible in the estimated mean value of displacement < v > (Fig.2d). A displacement mean of < v >≈ 0.11m means that the solution oscillates aroundthe beam equilibrium v0 ≈ 0.1042m, where v0 can be found from the minimum of thepotential energy (Eq. (2)) as

v0 =

√

g(mtN4 + ρAN9) − IEN6

2EIN7

, (8)

while any other value corresponds to cases with hopping (for σv > 0.25). The positive< v > mimics the asymmetry in the time series with respect to the horizontal reflectionthrough v = 0. This asymmetry is related to the limits of the simulation time, t ∈ [0 :3600], and the assumed initial conditions, v0 > 0.

To illustrate the system dynamics, examples of displacement and voltage time series,

6

(a)0 2000 4000 6000

−0.2

−0.1

0

0.1

0.2

time [s]

disp

lace

men

t v [m

]

(b)−0.2 −0.1 0 0.1 0.2

−0.02

−0.01

0

0.01

0.02

displacement v [m]

velo

city

v [m

/s]

(c)0 2000 4000 6000

−0.05

0

0.05

time [s]

Vol

tage

[V]

(d)0 2000 4000 6000

0

0.2

0.4

0.6

0.8

1

time [s]

num

ber

of h

ops

Figure 3: Simulation results for σF = 0.200N and tip mass mt = 0.038kg, with Gaussianwhite noise (ND) excitation Γ(t): displacement time series (a), phase portrait (b), voltagetime series (c), hops from the time simulation (d). Note that the number of hops is 0 asthe solution is localized around the buckled beam equilibrium v0.

displacement-velocity phase portraits, and the number of hops between potential wells,were plotted for the solutions with selected noise levels (for the case of ND) in Figs.3 to 5. In Fig. 3, σF = 0.200N and the solution was reduced to a single potentialwell oscillation (Fig. 3a-b). In contrast Figs. 4a-b (for σF = 0.320N) and 5a-b (forσF = 0.528N) clearly show the double well oscillations with a fairly large amplitude. Thecorresponding simulation time (t ∈ [0, 7200s] was longer than that used for the resultspresented in Fig. 2. The associated number of system hops is illustrated in Figs. 4d and5d. Note that the hops are intermittent for intermediate values of the excitation noiselevel σF = 0.320N (Fig. 4d). In the case of the stronger noise excitation σF = 0.528N(Fig. 5d) the hops appear to be fairly regular. As concluded from Fig. 2b, the outputpower strongly increases after passing σF = 0.25. This is also illustrated by the increasingvoltage output in Figs. 3c, 4c and 5c, where the corresponding U(t) time series are plottedfor σF = 0.200N, 0.320N, and 0.528N respectively. These results show an increase in thevoltage output.

7

(a)0 2000 4000 6000

−0.2

−0.1

0

0.1

0.2

time [s]

disp

lace

men

t v [m

]

(b)−0.2 −0.1 0 0.1 0.2

−0.02

−0.01

0

0.01

0.02

displacement v [m]

velo

city

v [m

/s]

(c)0 2000 4000 6000

−0.05

0

0.05

time [s]

Vol

tage

[V]

(d)0 2000 4000 6000

0

10

20

30

time [s]

num

ber

of h

ops

Figure 4: Simulation results for σF = 0.320N and tip mass mt = 0.038kg, with Gaussianwhite noise (ND) excitation Γ(t): displacement time series (a), phase portrait (b), voltagetime series (c) obtained through the single noise realisation. Fig. 4d presents the numberof hops between the potential wells. Note that for smaller σF (Fig. 3d) the number ofhops was 0.

5 ConclusionsWe have analyzed the piezoelectric harvester based on the elastic inverted beam me-

chanical resonator. Applying the stochastic kinematic excitation to the base we analyzedthe voltage output. By increasing the noise level we reported the appearance of thestochastic resonance phenomenon. The simulated results are characterized by relativelylarge variability (Fig 2a). As the results are averaged over ten different noise realizations,this variability occurs in spite of a single set of initial conditions used in all simulationsand shows the existence of multiple attractors in the examined system [13].

As expected from the previous studies [18], the effects of both kinds of noise (ND andUD) are very similar.

It should be noted that for realistic situations the parameters of the mechanical res-onator (for instance the tip mass value mt) can be adjusted to ensure the system isoperating in the vicinity of stochastic resonance. In that case the displacement signal toambient noise ratio has the highest value and the whole system is characterized by theoptimum performance. Our experiments on the system with stochastic excitation are inprogress and the results will be reported soon.

8

(a)0 2000 4000 6000

−0.2

−0.1

0

0.1

0.2

time [s]

disp

lace

men

t v [m

]

(b)−0.2 −0.1 0 0.1 0.2

−0.02

−0.01

0

0.01

0.02

displacement v [m]

velo

city

v [m

/s]

(c)0 2000 4000 6000

−0.05

0

0.05

time [s]

Vol

tage

[V]

(d)0 2000 4000 6000

0

20

40

60

80

100

time [s]

num

ber

of h

ops

Figure 5: Simulation results for σF = 0.528N and tip mass mt = 0.038kg, with Gaussianwhite (ND) noise excitation Γ(t): displacement time series (a), phase portrait (b), voltagetime series (c), hops from the time simulation (d).

AcknowledgementThe authors gratefully acknowledge the support of the Royal Society through Inter-

national Joint Project No. JP090343. MB and GL thank for the support of the 7thFramework Programme FP7-REGPOT-2009-1, under Grant Agreement No. 245479.

References[1] C.B. Williams, R.B. Yates, Analysis of a micro-electric generator for microsystems.

Sensors and Actuators A, 52, 8-11 (1996).

[2] S. Roundy, P. Wright, J. Rabaey, Energy Scavenging for Wireless Sensor Networks,Kluwer Academics Boston M.A. 2003.

[3] S.R. Anton, A. Erturk, D.J. Inmann, Multifunctional self-charging structures usingpiesoceramics and thin-film batteries, Smart Materials and Structures 19, 115021(2010).

[4] H.S. Kim, J.-H. Kim, J. Kim, A review of piezoelectric energy harvesting based onvibrations, Inter. J. Precision Engin. Manufacturing 12, 1129-1141 (2011).

9

[5] P.D. Mitcheson, E.M. Yeatman, G.K. Rao, A.S. Holmes, T.C. Green, Energy har-vesting from human and machine motion for wireless electronic devices, Proceedingsof the IEEE 96, 1457-1486.

[6] L. Tang,Y. Yang, C.K. Soh, Toward broadband vibration-based energy harvesting,Journal of Intelligent Material Systems and Structures 21, 1867-1897 (2010).

[7] A. Erturk, D. Inman, Piezoelectric Energy Harvesting, John Wiley & Sons Ltd.,Chichester, UK (2011).

[8] S. Priya, Advances in energy harvesting using low profile piezoelectric transducers,Journal of Electroceramics 19, 167-184 (2007).

[9] A. Erturk, J. Hoffmann, D.J. Inman, A piezomagnetoelastic structure for broadbandvibration energy harvesting, Applied Phys. Lett. 94 254102 (2009).

[10] E.S. Leland, P.K. Wright, Resonance tuning of piezoelectric vibrations energy scav-enging generators using compressive axial preload, Smart Mater. Struct. 15, 1413-1420 (2006)

[11] G. Litak, M.I. Friswell, S. Adhikari, Magnetopiezoelastic energy harvesting driven byrandom excitations, Applied Physics Letters, 96, 214103 (2010).

[12] C.A.K. Kwuimy, G. Litak, M. Borowiec, C. Nataraj Performance of a piezoelectricenergy harvester driven by air flow, Applied Physics Letters, 100, 024103 (2012).

[13] M.I. Friswell, S.F. Ali, S. Adhikari, A.W. Lees, O. Bilgen, G. Litak, Nonlinear piezo-electric vibration energy harvesting from an inverted cantilever beam with tip mass.J. Int. Mat. Syst. Struc. doi: 10.1177/1045389X12455722 (2012).

[14] F. Cottone, H. Vocca, L. Gammaitoni, Nonlinear energy harvesting, Phys. Rev. Lett.102, 080601 (2009).

[15] E. Halvorsen, Energy harvesters driven by broadband random vibrations, J. Mi-cromech. Syst. 17,1061-1071, (2008)

[16] L.-C.J. Blystad, E. Halvorsen, An energy harvester driven by colored noise, SmartMater. Struct. 20, 025011, (2011).

[17] M. Ferrari, M. Bau, M. Guizzetti, V. Ferrari, A single-magnet nonlinear piezoelec-tric converter for enhanced energy harvesting from random vibrations, Sensor andActuators A-Physical 172, 287-292 (2011).

[18] G. Litak, M. Borowiec, M.I. Friswell, S. Adhikari Energy harvesting in a magne-topiezoelastic system driven by random excitation with uniform and Gaussian dis-tributions. Journal Theor. Appl. Mech. 49, 757-764 (2011).

[19] N.A. Khovanova, I.A. Khovanov, The role of excitations statistic and nonlinearity inenergy harvesting from random impulsive excitations, App. Phys. Lett. 99, 144101(2011).

10

[20] M.F. Daqaq, Transduction of a bistable inductive generator driven by white andexponentially correlated Gaussian noise, J. Sound Vibr. 330, 2554-2564 (2011). [21,22, 23]

[21] J. Awrejcewicz, V.A. Krysko, On the vibration of the Euler-Bernoulli beam withclamped ends deflection constraints, Int. J. Bifur. Chaos, 1867-1878 (2005).

[22] I.V. Andrianov, J. Awrejcewicz, On the improved Kirchhoff equation modelling non-linear vibrations of beams, Acta Meccanica, 186, 2006, 135-139.

[23] J. J. Awrejcewicz, A.V. Krysko, V. Soldatov, V.A.Krysko, ’Analysis of the nonlineardynamics of the Timoshenko flexible beams using wavelets’, Journal of Computa-tional and Nonlinear Dynamics, 7, 011005 (2012).

[24] L.D. Zavodney, A.H. Nayfeh, The nonlinear response of a slender bean carryinglumped mass to a principal parametric excitation: theory and experiment. Int. J.Non-Linear Mech. 24, 105-125 (1989).

[25] E. Esmailzadeh, G. Nakhaie-Jazar, Periodic behavior of a cantilever beam with endmass subjected to harmonic base excitation, Int. J. Non-linear Mechanics, 33, 567-577(1998).

[26] L. Gammaitoni, P. Hanggi, P. Jung, F. Marchesoni, Stochastic resonance, Rev. Mod.Phys. 70, 223-287 (1998).

AppendixTo build the equations of motion (Eq. 4) we have to evaluate a number of integrals

in Eqs. (1) and (2). These can be easily calculated following [13] and also [24, 25], wherethe deflection model is assumed to be

ψ(x) = 1 − cos(πx

2L

)

,

After substituting the above expression into Eqs. (1) to (3), and some lengthly math-ematical manipulations [13, 24, 25], we obtain the constants N1 to N9 (see Eq. 4) as

N1 =

∫ L

0

ψ(x)2dx = L(3

2−

4

π

)

, N2 =

∫ L

0

ψ(x)dx =L

π(π − 2),

N3 =

∫ L

0

(

∫ x

0

ψ′(x)2dx)2

dx =π2

384L(2π2

− 9), N4 =

∫ L

0

ψ′(x)2dx =π2

8L,

N5 = ψ′(L) =π

2L,

N6 =

∫ L

0

ψ′′(x)2dx =π4

32L3, N7 =

∫ L

0

ψ′′(x)2ψ′(x)2dx =π6

512L5,

N8 =

∫ L

0

ψ′′(x)2ψ′(x)4dx =π8

4096L7, N9 =

∫ L

0

(

∫ x

0

ψ′(x)2dx)

dx =1

16(π2

− 4),

where ψ′(x) is the derivative dψ(x)/dx.

11