and Structures Non-linear piezoelectric vibration energy ...engweb.swan.ac.uk/~adhikaris/fulltext/journal/ft176.pdf · put for piezoelectric energy harvesting systems. Energy harvesting

Article

Journal of Intelligent Material Systemsand Structures23(13) 1505–1521� The Author(s) 2012Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1045389X12455722jim.sagepub.com

Non-linear piezoelectric vibrationenergy harvesting from a verticalcantilever beam with tip mass

Michael I Friswell1, S Faruque Ali1, Onur Bilgen1, Sondipon Adhikari1,Arthur W Lees1 and Grzegorz Litak2

AbstractA common energy harvesting device uses a piezoelectric patch on a cantilever beam with a tip mass. The usual configura-tion exploits the linear resonance of the system; this works well for harmonic excitation and when the natural frequencyis accurately tuned to the excitation frequency. A new configuration is proposed, consisting of a cantilever beam with atip mass that is mounted vertically and excited in the transverse direction at its base. This device is highly non-linear withtwo potential wells for large tip masses, when the beam is buckled. The system dynamics may include multiple solutionsand jumps between the potential wells, and these are exploited in the harvesting device. The electromechanical equa-tions of motion for this system are developed, and its response for a range of parameters is investigated using phase por-traits and bifurcation diagrams. The model is validated using an experimental device with three different tip masses,representing three interesting cases: a linear system; a low natural frequency, non-buckled beam; and a buckled beam.The most practical configuration seems to be the pre-buckled case, where the proposed system has a low natural fre-quency, a high level of harvested power and an increased bandwidth over a linear harvester.

Keywordsenergy harvesting, piezoelectric, non-linear dynamics

Introduction

Energy harvesting of ambient vibration is importantfor remote devices, for example, in structural healthmonitoring (Anton and Sodano, 2007; Beeby et al.,2006; Lefeuvre et al., 2005, 2006; Priya, 2007; Sodanoet al., 2004). Completely wireless sensor systems aredesirable, and this can only be accomplished by usingbatteries and/or harvested energy. Harvesting is attrac-tive because the energy generated can be used directlyor used to recharge batteries or other storage devices,which enhances battery life. Most of the results usingthe piezoelectric effect as the transduction method haveused cantilever beams and single frequency excitation,that is, resonance-based energy harvesting. The designof an energy harvesting device must be tailored to theambient energy available. For single frequency ambientexcitation, the resonant harvesting device is optimum,provided it is tuned to the excitation frequency. Ng andLiao (2005), duToit et al. (2005), Roundy (2005) andRenno et al. (2009) have proposed methods to optimisethe parameters of the system to maximise the harvestedenergy. Shu and Lien (2006a, 2006b) and Shu et al.

(2007) conducted a detailed analysis of the power out-put for piezoelectric energy harvesting systems.

Energy harvesting exploiting linear vibration hasbeen investigated widely, and explicit expressions foroptimal parameters are available in the literature (Erturkand Inman, 2011c). One of the drawbacks of linearenergy harvesters is that generally they are efficient onlywhen the excitation frequency is around the resonancefrequency (Daqaq, 2010). Therefore, most linear energyharvesting devices are designed on the assumption thatthe (base) excitation has some known form, typicallyharmonic excitation. However, there are many situationswhere energy harvesting devices are operating underunknown or random excitations, and in such situations,

1College of Engineering, Swansea University, UK2Department of Applied Mechanics, Lublin University of Technology,

Lublin, Poland

Corresponding author:

Michael I Friswell, College of Engineering, Swansea University, Singleton

Park, Swansea SA2 8PP, UK.

Email: [email protected]

harvesters with a broadband or adaptive response arelikely to be beneficial. One approach is to adaptivelychange the parameters of the linear harvester, so that itsnatural frequency becomes close to the excitation fre-quency as it changes (Wang et al., 2009). Such adaptivesystems may be difficult to implement in general andmay not adapt well to a broadband excitation. Ferrari etal. (2008) used an array of cantilever beam harvesterstuned to different frequencies.

An alternative approach to maximise the harvestedenergy over a wide range of excitation frequency usesnon-linear structural systems, and a range of deviceshave been proposed (Cottone et al., 2009; Gammaitoniet al., 2009, 2010). The key aspect of the non-linear har-vesters is the use of a double potential well function, sothat the device will have two equilibrium positions(Cottone et al., 2009; Ferrari et al., 2010; Mann andOwens, 2010; Quinn et al., 2011; Ramlan et al., 2010).Gammaitoni et al. (2009) and Masana and Daqaq(2011) highlighted the advantages of a double potentialwell for energy harvesting, particularly when inter welldynamics were excited. The simplest equation ofmotion with a double potential well is the well-knownDuffing oscillator, which has been extensively studied,particularly for sinusoidal excitation. The dynamics isoften complex, sometimes with coexisting periodic solu-tions and sometimes exhibiting a chaotic response. TheDuffing oscillator model has been used for manyenergy harvesting simulations, with the addition ofelectromechanical coupling for the harvesting circuit.One popular implementation of such a potential is apiezomagnetoelastic system based on the magnetoelas-tic structure that was first investigated by Moon andHolmes (1979) as a mechanical structure that exhibitsstrange attractor motions. Erturk et al. (2009) investi-gated the potential of this device for energy harvestingwhen the excitation is harmonic and demonstrated anorder of magnitude larger power output over the linearsystem (without magnets) for non-resonant excitation.One problem with multiple solutions to harmonic exci-tation is that the response can respond in the low-amplitude solution; Sebald et al. (2011) proposed amethod to excite the system to jump to the high ampli-tude solution at low energy cost. Stanton et al. (2010)and Erturk and Inman (2011b) investigated thedynamic response, including the chaotic response, forsuch a system. Cottone et al. (2009) used an invertedbeam with magnets and considered random excitation.Mann and Sims (2009) and Barton et al. (2010) used anelectromagnetic harvester with a cubic force non-linear-ity. Litak et al. (2010) and Ali et al. (2011) investigatednon-linear piezomagnetoelastic energy harvesting underrandom broadband excitation. McInnes et al. (2010)investigated the stochastic resonance phenomena for anon-linear system with a double potential well.

Another requirement of an energy harvester is toharvest reasonable amount of energy when the

excitation frequency is low. One example of this isenergy harvesting from vibration of long-span bridgesand tall buildings. A low-frequency piezoelastic orpiezomagnetoelastic harvester is difficult to realise dueto small physical dimensions of the devices. In this arti-cle, an inverted cantilever beam with piezoelastic patchloaded with a tip mass is investigated. The idea is toadjust the mass, such that the system is near bucklingand therefore has a low effective resonance frequency.The beam undergoes large deformations exhibiting anon-linear behaviour, and hence, geometric non-linearities are considered. By exploiting non-linearity,the aim is to have a low-frequency energy harvestingdevice that is relatively insensitive to a particular excita-tion frequency and responds with a relatively largeamplitude. This article reports theoretical, numericaland experimental investigation of the proposed device.

Inverted beam with tip mass

For non-linear energy harvesting, an inverted elasticbeam is considered with a tip mass, and the base is har-monically excited in the transverse direction. In thissection, we derive the governing equation of motionusing Euler–Bernoulli beam theory. The displacement–curvature relation of the beam is non-linear due to thelarge transverse displacement of the beam. We assumethat the thickness of the beam is small compared withthe length, so that the effects of shear deformation androtary inertia of the beam can be neglected. The beamis such that the first torsional resonance frequency ismuch higher than the excitation frequency, and thelumped mass is kept symmetric with respect to the cen-tre line. Hence, the vibration is purely planar, and weneglect the torsional modes of the beam in the analysis.These assumptions are consistent with the observationsin the laboratory.

Figure 1 shows the beam as a vertical cantilever oflength L with harmonic base excitation z(t)= z0 cosvt.The beam carries a concentrated tip mass, Mt, withmoment of inertia It, at a position Lt from the base ofthe beam. The horizontal and vertical elastic displace-ments at the tip mass are v and u, respectively, and s

represents the distance along the neutral axis of thebeam.

Consider an arbitrary point on the beam, P, at a dis-tance s from the base. This point undergoes a rigidbody translation due to the base excitation and a fur-ther displacement due to the elastic beam deformation,which is given by (vp(s, t), � up(s, t)). Hence, the pointP has undergone a total horizontal displacement ofz+ vp and a vertical displacement of �up. Let up(s, t)denote the rotation of the beam at s, and hence, therotation at the tip mass is u(t)=up(Lt, t), measured atthe mass centre.

In the following analysis, the beam is assumed tohave uniform inertia and stiffness properties; a non-

1506 Journal of Intelligent Material Systems and Structures 23(13)

uniform beam is easily modelled by including themechanical beam properties in the following energyintegrals. The beam has cross-sectional area A, massdensity r, equivalent Young’s modulus E and secondmoment of area I .

The kinetic energy of the beam-mass system is(Esmailzadeh and Nakhaie-Jazar, 1998)

T =1

2rA

ðL

0

_vp (s, t)+ _z� �2

+( _up (s, t))2h i

ds

+1

2Mt ( _v+ _z)2 + _u2� �

+1

2It

_f2 ð1Þ

where the translation of the tip mass is v(t)= vp(Lt, t)and u(t)= up(Lt, t) and the dot denotes differentiationwith respect to time. Equation (1) is obtained byneglecting the effect of rotary inertia of the beam mass.

The potential energy of the system is

P=1

2EI

ðL

0

(k(s, t))2ds� rAg

ðL

0

up(s, t)ds�Mtgu ð2Þ

where the curvature is (Ali and Padhi, 2009; Nayfehand Pai, 2004; Zavodney and Nayfeh, 1989)

k(s, t)=∂fp

∂s=f9p ð3Þ

where the prime denotes differentiation with respect tos and g is the gravitational constant. The slope of thebeam, fp, may be written in terms of the beam elasticdisplacement as

cosfp = 1� u9p or sinfp = v9p ð4Þ

Hence (Ali and Padhi, 2009; Nayfeh and Pai, 2004)

u9p = 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� v9p

2

q’

1

2v9p

2

or up(s, t)=1

2

ðs

0

(v9p(j, t))2dj ð5Þ

The second of equation (4) gives

fp(s, t)= sin�1 v9p’v9p +1

6v9p

3 ð6Þ

and differentiating this equation gives

k(s, t)=f9p =v99p

cosfp

=v99pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� v9p2

q ’v99p 1+1

2v9p

2

� �

ð7Þ

Equations (5) to (7) have been expanded as Taylorseries, and the terms in O(v4

p) and higher orders areneglected.

In this article, we assume that the tip mass is signifi-cantly larger than the beam mass, and hence, a single-mode approximation of the beam deformation is suffi-cient. The displacement at any point in the beam is rep-resented as a function of the tip mass displacementthrough a function for the beam deformation, c(s), as

vp(s, t)= vp(Lt, t)c(s)= v(t)c(s) ð8Þ

The displacement may be approximated by any func-tion satisfying the boundary conditions at s= 0, forexample (Esmailzadeh and Nakhaie-Jazar, 1998)

c(s)= lt 1� cosps

2L

ð9Þ

where lt is a constant, such that c(Lt)= 1, that is,lt = 1= 1� cos (pLt=2L)ð Þ. Similar results are obtained

P

z t z( ) = cos0 ωt

u

v

vp

up

Mt

s

Figure 1. Schematic representation of the inverted beamharvester. Mt denotes the tip mass attached to the elastic beam,while v and u denote the horizontal and vertical displacementsof the mass, respectively. Point P denotes an arbitrary point onthe beam whose position is described by the coordinates s, vp

and up. In this article, piezoelectric patches are placed along thebeam but are not shown here.

Friswell et al. 1507

using other displacement models, such as the staticbeam deflection. The implications of this assumptionfor the experimental validation will be discussed later.

Using this single-mode approximation, the kineticand potential energies of the system in terms of thetransverse displacement of the tip mass, v, are

T =1

2rA

ðL

0

_vc(s)+ _zð Þ2 + v _v

ðs

0

(c9(j))2dj

0@

1A

224

35ds

+1

2Mt ( _v+ _z)2 + v _v

ðLt

0

(c9(s))2ds

0@

1A

2264

375

+1

2It _vc9(Lt)+

1

2v2 _v(c9(Lt))

3

� �2

ð10Þ

=1

2rA N1 _v2 + 2N2 _v_z+ _z2 L+N3 v _vð Þ2h i

+1

2Mt ( _v+ _z)2 +N2

4 v _vð Þ2h i

+1

2It N5 _v+

1

2N 3

5 v2 _v

� �2

ð11Þ

and

P=1

2EI

ðL

0

vc(s)99+1

2v3(c9(s))2c99(s)

� �2

ds

� 1

2rAgv2

ðL

0

ðs

0

(c9(j))2dj

24

35ds

� 1

2Mtgv2

ðLt

0

(c9(s))2ds ð12Þ

=1

2EI N6v2 +N7v4 +

1

4N8v6

� �

� 1

2N9rAgv2 � 1

2N4Mtgv2 ð13Þ

Using the displacement model in equation (9), theconstants from N1 to N9 are given by

N1 =

ðL

0

(c(s))2ds= l2t

3p � 8

2p

� �L

N2 =

ðL

0

c(s)ds= lt

p � 2

p

� �L

N3 =

ðL

0

ðs

0

(c9(j))2dj

0@

1A

2

ds= l2t

p2(2p2 � 9)

384

� �1

L

N4 =

ðLt

0

(c9(s))2ds= l2t

p2

8

� �1

Lt

N5 =c9(Lt)= lt

p

2

1

Lt

N6 =

ðL

0

(c99(s))2ds= l2t

p4

32

� �1

L3

N7 =

ðL

0

(c9(s)c99(s))2ds= l4t

p6

29

� �1

L5

N8 =

ðL

0

(c9(s))4(c99(s))2ds= l6t

p8

4096

� �1

L7

N9 =

ðL

0

ðs

0

(c9(j))2dj

24

35ds= l2

t �1

4+

1

16p2

� �

ð14Þ

Different displacement models will lead to differentconstants from N1 to N9, which may be easily derivedusing the displacement function c(s).

The equation of motion of the beam-mass system isderived in terms of the displacement of the tip massusing Lagrange’s equations as

N 25 It +Mt + rAN1 + rAN3 +MtN

24 +N 4

5 It

� �v2

� �€v

+ rAN3 +MtN24 +N4

5 It

� �v _v2

+ EIN6 � N9rAg � N4Mtg + 2EIN7v2� �

v

= � rAN2 +Mt½ �€z ð15Þ

Damping may also be added to these equations ofmotion, for example, viscous, material or aerodynamicdamping.

Equilibrium positions

The equilibrium positions with no forcing are obtainedby setting the velocity and acceleration terms to zero inequation (15) to give

EIN6 � N9rAg � N4Mtg + 2EIN7v2� �

v= 0 ð16Þ

This equation has either one or three solutions, andv= 0 is always a solution. Since N4.0, there are threesolutions if

Mt.EIN6 � N9rAg

N4g=Mtb ð17Þ

where Mtb is the tip mass so that the beam is about tobuckle. If the beam mass is neglected, this gives theEuler buckling load as


Mtbg =EIN6

N4

=EIp2

4L2ð18Þ

If equation (17) is satisfied, then the non-zero equili-brium positions are given by

v0b =6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN9rAg +N4Mtg � EIN6

2EIN7

rð19Þ

For perturbations about the equilibrium solution atv= 0, the linearised equation of motion for the freeresponse is

N 25 It +Mt + rAN1

� �€v+ EIN6 � N9rAg � N4Mtg½ �v= 0

ð20Þ

showing that the v= 0 equilibrium position is unstableafter buckling (Mt.Mtb) and that before buckling(Mt\Mtb) the natural frequency for small vibrations isgiven by

v2n =

EIN6 � N9rAg � N4Mtg

N25 It +Mt + rAN1

ð21Þ

After buckling the linearised equation of motionabout the equilibrium position v0b becomes, using equa-tion (19)

N25 It +Mt + rAN1 + rAN3 +MtN

24 +N 4

5 It

� �v2

0b

� �€h

+ 4EIN7v20bh= 0 ð19Þ

where v= v0b +h. Hence, the natural frequenciesabout both buckled equilibrium positions are

v2nb =

4EIN7v20


24 +N4

5 It

� �v2

0b

ð23Þ

Coupled electromechanical model

There has been a significant modelling effort of piezo-electric materials as distributed transducers and manyreview articles have been published (see, for example,Benjeddou, 2000; Chee et al., 1998; Chopra, 2002;Crawley, 1994; Leo, 2007). The analyses range fromsimple devices such as uniform beams and plates in lin-ear dynamics to more complicated configurations suchas composites under non-linear and non-uniform load-ing and dynamics (such as helicopter blades and air-craft wings). A majority of the research deals with themodelling of symmetric (bimorph) beams and plates;however, this article will also consider asymmetric(unimorph) beams. The symmetric device theoreticallyproduces only bending strains; in contrast, the asym-metric device has bending–extension coupling. Thereare two popular sets of assumptions for modellingstrain-induced actuation and sensing. First, the

uniform strain model assumes that the through-the-thickness variation of strain in the active piezoelectricdevice is uniform. This assumption holds true for caseswhere the passive substrate material is relatively thickcompared to the active material. The second caseallows for the linear variation of strain in the activematerial and follows the assumptions of the Euler–Bernoulli model. Crawley and De Luis (1987) andCrawley and Anderson (1990) gave the uniform strainand Euler–Bernoulli derivations for strain-inducedactuation. They demonstrated several important resultssuch as the increased effectiveness of the induced-strainactuators for stiffer and thinner bonding layers. Erturkand Inman (2011a) and Leo (2007) give further detailsof the modelling of piezoelectric sensors and actuatorsintegrated with the beam structures.

Suppose that piezoelectric layers added to a beam ineither a unimorph or a bimorph configuration. Thenthe moment about the beam neutral axis produced by avoltage V across the piezoelectric layers (Crawley andAnderson, 1990; Crawley and De Luis, 1987) may bewritten as

ML(s, t)= gcV (t) ð24Þ

where the constant gc depends on the geometry, config-uration and piezoelectric device.

Hence, for a bimorph with piezoelectric layers in the31 configuration, with thickness hc, width bc and con-nected in parallel

gc =Ed31bc h+ hcð Þ ð25Þ

where h is the thickness of the beam and d31 is thepiezoelectric constant. For a unimorph, the constant is

gc =Ed31bc h+hc

2� �z

� �ð26Þ

where �z is the effective neutral axis (Park et al., 1996).These expressions assume a monolithic piezoceramicactuator perfectly bonded to the beam; Bilgen et al.(2010) considered the effect of the structure of a Macro-Fiber Composite (MFC) on the coupling coefficientand the effect of the bond and Kapton layers. Themechanical stiffness and mass density of the piezoelec-tric layers should also be included in the beam constantsalready derived.

The work done by the piezoelectric patches in mov-ing or extracting the electrical charge is

W =

ðLc

0

ML(s, t)k(s)ds ð27Þ

where Lc is the active length of the piezoelectric mate-rial, which is assumed to be attached at the clampedend of the beam. Using the approximation for k in


equation (7), and the displacement model in equation(8), we have

W’ Q1v+1

3Q2v3

� �V ð28Þ

where

Q1 = gc

ðLc

0

c99(s)ds= gcc9(Lc) ð29Þ

and

Q2 = 3gc

ðLc

0

1

2c99(s)(c9(s))2ds=

1

2gc(c9(Lc))

3 ð30Þ

Equation (28) results in additional terms in themechanical equation of motion, which becomes


24 +N 4

5 It

� �v2

� �€v

+ rAN3 +MtN24 +N4

5 It

� �v _v2

+ EIN6 � N9rAg � N4Mtg + 2EIN7v2� �

v

�Q1V �Q2v2V = � rAN2 +Mt½ �€z ð31Þ

On the electrical side, the piezoelectric patches maybe considered as a capacitor, and the charge they pro-duce is given by Q1v+Q2v3, where Q1 and Q2 aregiven by equations (29) and (30), respectively. The elec-trical circuit considered is represented by a resistive

shunt connected across the piezoelectric patch. Theelectrical equation then becomes

Cp_V +

V

Rl

+Q1 _v+Q2v2 _v= 0 ð32Þ

where Rl is the load resistor and Cp is the capacitance ofthe piezoelectric patch.

The average power scavenged between times T1 andT2 is calculated as

Pave =1

T2 � T1

ðT2

T1

V (t)2

Rl

dt ð33Þ

Numerical simulations

The parameters considered for the numerical simula-tions are given in Table 1. The beam-mass system isexcited at the base with harmonic excitation. Note thatwhen the tip mass is changed, the ratio of Mt=It is main-tained; this is equivalent to increasing the tip masswidth to increase the tip mass.

Figure 2(a) shows the equilibrium position of the tipmass, using the analysis described in section‘Equilibrium positions’, and shows that the post-buckled response has two equilibrium positions. Figure2(b) shows the corresponding natural frequency of thelinearised system with the change in the tip mass; boththe pre-buckled and post-buckled natural frequenciesare given. Linearisation about both equilibrium posi-tions provides the same natural frequencies as the sys-tem is assumed to be symmetric. Figure 2(b) shows thatthe natural frequency of the inverted elastic pendulumdecreases with increasing tip mass and is zero at theEuler buckling load corresponding to an estimated tipmass of 10:0 g. Further increases in tip mass cause thebeam to buckle, and the natural frequencies about thestable equilibrium positions increase with the tip mass.Thus, the inverted elastic beam-mass system is able toresonate at low frequencies close to the buckling condi-tion. The post-buckled equilibrium positions are quite

0 5 10 15 20

−200

−100

0

100

200

Tip mass (g)

Equ

ilibr

ium

pos

ition

(m

m)

0 5 10 15 200

0.5

1

1.5

2

Tip mass (g)

Nat

ural

fre

quen

cy (

Hz)

(a) (b)

Figure 2. The effect of the tip mass on (a) the equilibrium position and (b) the corresponding natural frequencies for the stableequilibrium positions. The dashed line denotes unstable equilibrium positions.

Table 1. Parameter values used in the simulation.

Beam and tip mass Energy harvester

r 7850 kg/m3 Lc 28 mmE 210 GN/m2 bc 14 mmb 16 mm hc 300 mmh 0.254 mm gc �4:00310�5 Nm=VL= Lt 300 mm Cp 51.4 nFIt=Mt 40.87 mm2 Rl 105 � 108 O


sensitive to the tip mass, and in the simulation study, atip mass of 10:5 g was used, unless stated otherwise.

Figure 3 shows the time history of the tip displace-ment and the phase portrait of the inverted beamenergy harvester system with the parameters given inTable 1 and with zero initial displacement and velocity.The response is only shown from t = 1300 s tot = 1500 s to demonstrate the characteristics of thebeam dynamics. In particular, the beam response some-times oscillates around one of the equilibrium positionsand sometimes exhibits large oscillations either side ofthe unstable zero displacement position. The corre-sponding voltage across the piezoelectric layer is shownin Figure 4. Figures 3(a) and 4 show that when the tipmass oscillates close to one of the equilibrium posi-tions, then the voltage across the piezoelectric layer(and therefore the power scavenged) is less than whenthe tip mass moves back and forth between the poten-tial wells.

The value of the tip mass is now swept from 8 to 20g. The time response is simulated, with zero initial

displacement and velocity, for 8000 cycles to ensurethat the transient dynamics have decayed. For the last100 cycles, the Poincare points are sampled, and thepoints corresponding to the displacement response ofthe tip mass are shown in Figure 5(a) as a bifurcationdiagram (Nayfeh and Mook, 1979). The bifurcationdiagram is complex and shows periodic solutions at theexcitation frequency (a single dot) at some values of tipmass, periodic responses with a period larger than thatof the excitation (multiple dots) and chaotic responses(solid blocks of dots). Examples of these responses willbe shown later. For the highly buckled case (for largetip masses), the solution is often periodic although thecharacter of the solutions changes abruptly as the tipmass changes. For these cases, there are likely to bemultiple coexisting solutions, and the solutionsobtained in the bifurcation diagram arise from the ini-tial conditions chosen. The asymmetry in the responseabout the vertical beam position arises because of thephasing of the forcing, which determines the timeswhen the Poincare points are sampled. A discontinuityis seen at Mt = 8:8 g, which corresponds to the jump inthe resonance due to the hardening stiffness character-istic. This looks different from the standard jump phe-nomena because here the resonance frequency is variedwhile keeping the excitation frequency fixed; normally,the excitation is changed for a fixed system.

Figure 5(b) shows the average power scavenged bythe piezoelectric patches, and this also shows a peak atMt = 8:8 g due to the resonance. The power scavengedat other values of the tip mass in the pre-buckled caseis very low. The average power scavenged in the post-buckled regime shows various trends that highlight theexistence of multiple coexisting solutions. This is shownmost clearly at higher values of tip mass where low-energy solutions exist where the beam response is solelywithin one potential well and high-energy solutionsexist where the beam response oscillates between bothpotential wells. For tip masses just higher than thebuckling mass, the response may be either periodic or

1300 1350 1400 1450 1500

−100

−50

0

50

100

Time (s)

Tip

mas

s di

spla

cem

ent (

mm

)

−100 −50 0 50 100

−200

−100

0

100

200

Displacement (mm)

Vel

ocity

(m

m/s

)

(a) (b)

Figure 3. Harvester non-periodic response for the parameters given in Table 1 and a harmonic excitation with z0 = 16 mm atfrequency 0:5 Hz: (a) displacement time history and (b) phase portrait for the tip mass. The dashed horizontal lines in (a) show theequilibrium positions of the tip mass. The dots in (b) represent the Poincare points. The response was obtained using zero initialconditions for the tip mass displacement and velocity.

1300 1350 1400 1450 1500

−500

0

500

Time (s)

Vol

tage

(m

V)

Figure 4. Time history of the voltage across the piezoelectriclayers with the parameters given in Table 1 and correspondingto the response given in Figure 3.


chaotic, and the chaotic response often gives a slightlyhigher average power output.

The response dynamics is further investigated inFigure 6 through the detailed analysis of time responsesfor typical tip mass values. Figure 6(a) shows period 1response at the pre-buckled resonance. Figure 6(b) alsoshows a period 1 response, but the beam is in the post-buckled regime where most of the responses are chaoticand produce lower power output. An example of thechaotic response is given in Figure 6(c). Figure 6(d)shows the response for the highest average outputpower, which shows some chaotic response superim-posed on a period 1 response. Examples of low- andhigh-power solutions for large values of tip mass areshown in Figure 6(e) and (f). The low power responseoscillates in one of the potential wells, whereas thehigh-power response is a period 7 response crossingback and forth between the potential wells. It shouldbe emphasised that other solutions also exist for othervalues of tip mass, for example, period 3 or period 5solutions.

Figure 7 shows a parametric study varying the resis-tance across the piezoelectric layer. Figure 7(a) shows abifurcation diagram and highlights that periodicresponses are seen at high resistance. Figure 7(b) showsthe corresponding average power output and shows aclear peak for the period 3 oscillations close toRl = 10MO. This optimum load resistance is very highbecause of the low capacitance of the piezoelectriclayers and because the excitation frequency is very low.

The effect of the base excitation amplitude and fre-quency is shown in Figures 8 and 9, respectively. At lowexcitation amplitudes, the beam mass vibrates withinone of the potential wells, and the power available isvery low. As the excitation amplitude is increased, thetip mass starts jumping from one potential well to theother, which increases the power harvested from thesystem. For higher excitation amplitudes, the response

can be either chaotic or periodic, and the periodic solu-tions with motion across both the potential wells have ahigher power output. At very high excitation ampli-tudes, the response is period 1, and this gives a higherpower output.

For excitation frequencies up to about 0.5 Hz, theresponse is predominantly periodic with an increasingresponse amplitude and corresponding power output.For higher frequencies, the response becomes chaotic,before becoming periodic (with various periods) for fre-quencies above about 0.56 Hz. The power output inFigure 9(b) shows that the responses have differentcharacteristics, and again, multiple coexisting solutionsare likely to occur for these excitation frequencies.

Experimental testing

An experimental system was built to validate the modeland to determine the effects of unmodelled non-linearities (e.g. piezoelectric hysteresis, noise in the baseexcitation and out-of-plane motion). As shown in themodel earlier, the main parameters of interest are thetip mass, the load resistance and the base excitationamplitude and frequency. The response of the beam ischaracterised by its transverse displacement and poweroutput through a resistive shunt. A load resistor is usedto characterise the energy harvesting performanceinstead of other passive or active circuits; therefore, theanalysis is focused on the fundamental non-lineardynamic behaviour of the cantilever energy harvester.

A National Instruments (NI) cDAQ 9172 dataacquisition system, controlled with a code written inLabVIEW software, was employed to automaticallyexamine the mechanical and electrical responses of theinverted cantilever piezocomposite beam. For eachexperiment, a tip mass, constructed using several mag-nets, was selected and attached to the beam manually.A load resistor was then selected using an

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Figure 5. The effect of the variation of the tip mass for a base excitation of z0 = 16 mm at frequency 0:5 Hz and for a loadresistance Rl = 100 kO: (a) bifurcation diagram and (b) average harvested power. The results were obtained using zero initialconditions for the tip mass displacement and velocity.


electromagnetic relay circuit, and the base displacementpeak amplitude was chosen. The frequency of excita-tion was then swept in both increasing and decreasingdirections. This process was repeated for all of theselected parameters.

The beam was excited for 30 complete cycles (ofbase excitation) for each combination of parameters in

order to minimise the effect of transient motion. Onlythe last 10 cycles were recorded and analysed. Since thefrequency was incremented in small steps and the wave-form was continuous between each frequency, the dis-turbance (e.g. rapid accelerations) to the beam wasminimised. This is important since such disturbanceswill result in a premature transition to an alternative

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Figure 6. Phase portraits of the tip mass response for a base excitation of z0 = 16 mm at frequency 0:5 Hz and for a loadresistance Rl = 100 kO: (a) Mt = 8:76 g, Pave = 7:65 mW, (b) Mt = 10:39 g, Pave = 16:3 mW, (c) Mt = 10:5 g, Pave = 2:40 mW, (d)Mt = 10:61 g, Pave = 17:8 mW, (e) Mt = 18:6 g, Pave = 0:056 mW and (f) Mt = 18:65 g, Pave = 10:2 mW. The dots represent thePoincare points. The results were obtained using zero initial conditions for the tip mass displacement and velocity.


solution as the frequency is increased and decreased.Although most ambient sources of vibration will bemulti-tone and have noise, the current research concen-trates on single-tone excitations.

The control signal for the base excitation was pro-duced by an NI 9263 cDAQ module with 16 bit resolu-tion (set to 610V range) at a generation rate of 10kHz. This control signal was low-pass (LP) filteredusing a Kemo (type VBF/24) elliptic filter with a 5-Hzcut-off frequency to minimise the high-frequency noisefrom the digital-to-analog converter (DAC). Note thatthe filtered output of the DAC was not measured asthe reference signal; therefore, the lag effect of the filterwas avoided. The base excitation signal was connectedto a Bytronic Pendulum Control System, consistedof a belt-driven linear slider that moves on a track and

is actuated by a direct current (DC) motor. A multi-turn potentiometer monitors the position of the beltand hence the position of the linear slider. A displace-ment feedback controller ensures that the displacementis proportional to the control signal. The linear sliderhas low inertia and that there is no return spring (as inan electromagnetic shaker). The fact that the slider haslow inertia means that the base motion is affected bysmall imperfections in the linear track. In addition,since the system is driven by the forcing of the DCmotor only (e.g. no return spring), the actual excitationdeviates from the desired harmonic excitation near the_z= 0 condition. Both of these deviations are measuredby the potentiometer; however, their effects on the gen-eral motion and the power output were assumednegligible.

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Figure 8. The effect of the amplitude of the base excitation at an excitation frequency of 0:5 Hz, a tip mass of Mt = 10:5 g and aload resistance Rl = 100 kO: (a) bifurcation diagram and (b) average harvested power. The results were obtained using zero initialconditions for the tip mass displacement and velocity.

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Figure 7. The effect of the resistance across the piezoelectric layer, for a tip mass of Mt = 10:5 g and a base excitation of amplitudez0 = 16 mm at frequency v= 0:5 Hz: (a) bifurcation diagram and (b) average harvested power. The results were obtained using zeroinitial conditions for the tip mass displacement and velocity.


A clamping mechanism, which applies uniform pres-sure across the clamped surface of the beam, wasattached to the linear slider. All experiments presentedin this article were conducted without removing thebeam from the clamp; therefore, consistent boundaryconditions were achieved across all cases. A beam madeof spring steel of thickness 0:245mm, width 15:88mmand free length 293mm was used as the cantilever.Figure 10 shows the beam mounted on the linear slider.

A single piezocomposite patch, the MFC modelM2814-P2 manufactured by Smart Material Corp., ofactive length 28 mm and active width 14 mm wasbonded to the beam near the clamped end in a unim-orph configuration. The MFC was developed at theNASA Langley Research Center (Wilkie et al., 2000).An MFC is a flexible, planar actuation device thatemploys rectangular cross-section, unidirectional piezo-ceramic fibres (PZT 5A) embedded in a thermosettingpolymer matrix (High and Wilkie, 2003). An electrome-chanical characterisation of the mechanical and piezo-electric behaviour of the MFC device can be found inBilgen et al. (2012). Since the purpose here is to improvepower output, the MFC with through-the-thicknesspoling (type P2), which operates in the 31 electromecha-nical mode, was chosen. The 31-mode device hasapproximately 40 times higher capacitance compared tothe interdigitated 33-mode device (type P1). The patchwas aligned to the beam symmetrically in the widthwisedirection and as close to the base as possible. The MFCwas bonded to the beam using a 3M DP460 type two-part epoxy and let for cure under ’1 atm vacuum.

The tip mass was implemented using several disc-likeneodymium magnets with diameter 10 mm, height 5mm and approximate mass 1:75 g, whose positionscould be moved easily. In the model, the mass wasassumed to be at the end of the beam; the portion of

the beam above the magnets in the experiment has littleeffect on the system dynamics.

The signals of interest are measured using a NI 9215analog-to-digital converter (ADC) module with 16 bitresolution (set to 610V range) at a variable samplerate. During the acquisition period, 100 points per baseexcitation cycle were captured. Three signals were mea-sured: the potentiometer output, which is proportionalto the base displacement; the laser displacement sensor(MTI LTC-300-200-SA with 620 mm resolution),which measures the mid-line of the beam at 100 mmfrom the base; and the voltage output of the MFCpiezocomposite device. A 10:1 voltage divider probe

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Figure 9. The effect of the frequency of the base excitation for an excitation amplitude of z0 = 16 mm, a tip mass of Mt = 10:5 gand a load resistance Rl = 100 kO: (a) bifurcation diagram and (b) average harvested power. The results were obtained using zeroinitial conditions for the tip mass displacement and velocity.

Figure 10. Picture of the experimental set-up. (a) Linear sliderand the inverted cantilever beam with 14:0 g tip mass. (b) Baseof the beam showing the MFC device. (c) Tip mass of 10:5 gshown nearly vertical at the stable equilibrium. (d) Tip mass of14:0 g showing approximately 458 end slope in a stableequilibrium. Note that laser displacement sensor is not shown.MFC: Macro-Fiber Composite.


(Agilent N2862A) with equivalent 9:95MO input impe-dance is used to monitor the voltage output of thepiezocomposite device.

As noted earlier, four parameters were varied. Thetip mass values were chosen as Mt = 0, 10:5 g (six mag-nets, Lt = 287mm, pre-buckling) and 14 g (eight mag-nets, Lt = 284mm, post-buckling) representing threefundamentally different dynamic behaviours. A total of10 load resistance values were applied. Table 2 showsthe effective load resistances connected to the MFCdevice. Note that the probe and ADC input impedanceswere included in the effective load resistance as well asthe selected resistor. The load resistances were mea-sured using an Agilent digital multimeter.

Base displacement peak amplitudes of 5, 10, 15, 20

and 25mm were utilised. This range was not exceededdue to several reasons, which are artificially introducedby the experimental set-up. First, large beam displace-ments exceed the measurement range of the laser,although the electrical response could still be measured.

Second, the magnets attract and attach to the slidertrack when high curvatures are present. In addition tothe parameters above, the frequency of base excitationwas examined in the range 0.3–3 Hz, although therange was adjusted depending on the tip mass.

Experimental results

As noted earlier, three tip mass values are evaluated. Inthis section, the fundamental response of the vibrationenergy harvester is evaluated in terms of two quantities:the measured displacement at a single point 100 mmfrom the base of the beam divided by the base displace-ment, and the measured power output through theknown effective resistance divided by the base displace-ment squared. It is important to note that the responsewaveforms are not necessarily a single harmonic; there-fore, each waveform is integrated over the time of inter-est to calculate their average values. The average valueof the beam displacement and power output is obtainedby integrating the measured values of the waveformover 10 cycles and dividing the integral by the totaltime. In the case of base displacement, the absolute val-ues are taken before the integration.

Figure 11(a) shows the ratio of the average displace-ment measured at the laser to the harmonic displace-ment response at the base of the inverted cantileverbeam with no tip mass. As expected, the response isapproximately linear, with a natural frequency range of2.43–2.47 Hz for the base displacement range of 5–20mm. As the base displacement amplitude is increased,the mechanical output/input ratio decreases and thepeak frequency increases. These results are typical for ahardening non-linearity. Figure 11(b) shows the ratio ofthe average power output to the square of the harmonic

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Figure 11. The experimental results for a tip mass of Mt = 0 g and a base excitation with amplitudes of z0 = 5 (solid), 10 (dashed),15 (dash–dot) and 20 mm (dotted) at a range of frequencies: (a) response measured by laser and (b) average harvested power.Rl = 1:658 MO for all measurements.

Table 2. Resistor values used in the experiment.

Nominal resistance Measured resistanceat piezo output

10 kO 9.91 kO100 kO 99.3 kO150 kO 146.9 kO330 kO 318.3 kO450 kO 450.6 kO1 MO 910 kO1.5 MO 1.658 MO2.2 MO 2.272 MO3.3 MO 3.344 MOOpen 9.944 MO


displacement response at the base of the inverted canti-lever beam with no tip mass. If the system was linear,then these curves would overlay for different values ofbase excitation; however, in this case, the ratio reducesas the base displacement amplitude increases.

Figure 12 shows the average displacement and powerresults for the inverted cantilever beam with a tip massof 10:5 g. At this value of tip mass, the beam has notbuckled, and the simulations show that the maximumpower is generated before the beam buckles, close tothe resonance. This prediction is consistent with theexperimental response when pre-buckled and post-buckled cases are compared. The resonance frequencyof the device can be easily tuned by moving the tipmass, and hence, the resonance may be tuned to theharmonic excitation frequency. The responses show theclassical jump phenomena for a hardening non-linear-ity, where the jump down occurs when slowly increasingthe excitation frequency and the jump up occurs whenreducing the excitation frequency. Thus, there is a rangeof frequencies where two stable coexisting solutionsexist.

Figure 13 shows the effect of varying the load resis-tance on the maximum power output for the invertedcantilever beam with a tip mass of 10.5 g. For eachload resistance and base excitation amplitude, the exci-tation frequency is swept in both the increasing and thedecreasing directions. Since the beam represents a hard-ening non-linearity, the maximum power occurs duringthe upward frequency sweep. The maximum power isidentified during this sweep and included in Figure 13.There is an optimum load resistance that gives the max-imum power (compare to Figure 7(b)), and the opti-mum resistance value decreases with increasing baseexcitation amplitude. This is expected as the maximumresponse occurs at a higher frequency when the

excitation amplitude is increased. The range of loadresistance values is very high, because the frequenciesof interest are very low. Also, note that the maximumresistance that may be implemented is limited by theADC input impedance.

Figure 14 shows the average displacement and powerresults for the inverted cantilever beam with a tip massof 14 g. At this tip mass value, the beam is buckled, andthe dominant mode of vibration is a single-well oscilla-tion. When base displacement amplitude is sufficiently

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Figure 12. The experimental results for a tip mass of Mt = 10:5 g and a base excitation with amplitudes of z0 = 5 (solid), 10(dashed), 15 (dash–dot) and 20 mm (dotted) at a range of frequencies: (a) response measured by laser and (b) average harvestedpower. Rl = 9:944 MO for z0 = 5 and 10 mm and Rl = 3:344 MO for z0 = 15 and 20 mm.

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Figure 13. The experimental maximum power for a tip massof Mt = 10:5 g and a base excitation with amplitudes of z0 = 5(solid), 10 (dashed), 15 (dash-dot) and 20 mm (dotted) for arange of load resistance.


large, or a resonance is excited about a specific stableequilibrium, cross-well oscillations may occur, althoughnone were observed in Figure 14. The power output forthe post-buckled configuration is significantly smallerthan that for the pre-buckled configuration.

Simulated results

The experimental system was modelled using the equa-tions of motion of the inverted cantilever beam devel-oped earlier in order to validate the model. There aretwo linked issues with simulating the tested system: theparameters of the model may only be obtained approx-imately from the geometry of the structure and thematerial properties, and the response is highly sensitiveto some parameters in the model. In addition, dampingis very difficult to model and in reality will mainly con-sist of viscoelastic material damping and air damping.In the model, a linear viscous damper has beenincluded, where the coefficient varies with the responseamplitude.

The single piezoelectric patch adds significant stiff-ness to the beam and will also cause a shift in the neu-tral axis of the beam. This will also change the modeshapes of the beam and therefore the displacementfunction if taken as the first mode. Here, we have con-sidered only a very simple model of the beam, wherethe displacement model is given by equation (9) andthe neutral axis is assumed to remain at the centre ofsteel beam. Using the mechanical properties and geo-metry of the MFC gives an increase in EI of 100% overand above that for the uniform steel beam, and a corre-sponding increase in the N6 and N7 terms in the equa-tions of motion of 19% and 0.57%, respectively. Theelectromechanical coupling coefficient is calculated asgc = � 1:49310�5 Nm=V.

The viscous damping coefficient is adjusted for eachsimulated run to give a similar qualitative response tothe measured response. The damping from the air resis-tance is likely to increase with higher amplitudemotion, and hence, the coefficient will increase for highexcitation amplitudes. For the cases without the tipmass and when Mt = 10:5 g, the viscous damping coef-ficient was set at 0:0032, 0:0038, 0:0045 and 0:0053 forexcitation amplitudes of z0 = 5, 10, 15 and 20mm,respectively. When Mt = 14 g, the coefficient wasincreased to 0:0048, 0:0057, 0:009 and 0:014 forz0 = 5, 10, 15 and 20mm, respectively. Note that thesimulated displacements are given at the tip mass,whereas the experimental displacements are given at afixed height determined by the position of the lasersensor.

Figures 15 to 17 show the simulated results corre-sponding to the measurements given in Figures 11 to 14and show similar trends to the experimental results.Without a tip mass, Figure 15 shows that the responseis approximately linear with the hardening non-linearityslightly increasing the natural frequency and slightlyreducing the frequency response function amplitude, asthe base excitation amplitude increases. When a tipmass of Mt = 10:5 g is added, the simulated results inFigure 16 show the jump phenomena of a hardeningnon-linearity. The jump frequencies are very sensitiveto the damping coefficient and the tip mass (comparedto that required for the beam to buckle).

For a tip mass of Mt = 14 g, the beam has buckled,and this results in a softening non-linearity with somejumps, with the response located in a single potentialwell, as shown in Figure 17. In this case, the dampingcoefficients have had to be increased; with lower damp-ing, at some frequencies, the beam can hop betweenpotential wells, and multiple solutions coexist.

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Figure 14. The experimental results for a tip mass of Mt = 14 g and a base excitation with amplitudes of z0 = 5 (solid), 10(dashed), 15 (dash-dot) and 25 mm (dotted) at a range of frequencies: (a) response measured by laser and (b) average harvestedpower. Rl = 9:944 MO for all measurements.


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Figure 16. The simulated results for a tip mass of Mt = 10:5 g and a base excitation with amplitudes of z0 = 5 (solid), 10 (dashed),15 (dash–dot) and 20 mm (dotted) at a range of frequencies: (a) tip mass response and (b) average harvested power. Rl = 9:944 MOfor z0 = 5 and 10 mm and Rl = 3:344 MO for z0 = 15 and 20 mm.

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Figure 15. The simulated results for a tip mass of Mt = 0 g and a base excitation with amplitudes of z0 = 5 (solid), 10 (dashed), 15(dash–dot) and 20 mm (dotted) at a range of frequencies: (a) tip mass response and (b) average harvested power. Rl = 1:658 MO forall cases.

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Figure 17. The simulated results for a tip mass of Mt = 14 g and a base excitation with amplitudes of z0 = 5 (solid), 10 (dashed),15 (dash-dot) and 20 mm (dotted) at a range of frequencies: (a) tip mass response and (b) average harvested power. Rl = 9:944 MOfor all cases.


Subharmonic resonances are also clear in the simulated

results, and the resonance at twice the primary reso-

nance frequency is readily identified. Subharmonic

resonances are not apparent on the experimental results

although there is some increase in amplitude at these

higher frequencies. The power generated in this simu-

lated case is also very low because the response stays

completely within in a single potential well. The equili-

brium configuration and the dynamic response are very

sensitive to all of the inertia and stiffness parameters in

the equations of motion, in addition to the damping

properties, and this makes a quantitative comparison

with the experimental results difficult.

Conclusion

The proposed energy harvesting system addresses avery difficult problem where energy is required from astructure with low excitation frequency and high dis-placement, such as a highway bridge. A resonant linearharvester based on a cantilever beam is difficult toimplement because the low natural frequency requiresa very large or a very flexible beam. In this article, alow-frequency piezoelectric energy harvester is pro-posed using an inverted elastic beam-mass system. Theequations of motion for the proposed system weredeveloped, the response was simulated and this modelwas validated experimentally. The results show that theharvester has the potential to scavenge power depend-ing on the proper choice of the tip mass and otherparameters. In particular, choosing a tip mass so thatthe beam is almost buckled gives a relative bandwidth(defined using the half power points) up to twice thatof the linear harvester. The maximum power harvestedis also significantly greater, once the lower excitationfrequencies are accounted for. If the beam is buckled,then the system exhibits the common non-linear systemcharacteristics such as coexisting solution, includingchaotic responses. In the buckled configuration, signifi-cant power is only harvested if the excitation is suffi-cient for the system to hop between the potential wellsand hence give a large displacement response.

Funding

This study was supported by the Royal Society throughInternational Joint Project No. HP090343. Dr Ali receivedfunding from the Royal Society through a NewtonFellowship. Prof. Adhikari received the support from theRoyal Society through a Wolfson Research Merit Award.

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