An energy harvester using piezoelectric cantilever beams undergoing coupled

bending–torsion vibrations

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Smart Mater. Struct. 20 (2011) 115007 (11pp) doi:10.1088/0964-1726/20/11/115007

An energy harvester using piezoelectriccantilever beams undergoing coupledbending–torsion vibrations

A Abdelkefi1,2, F Najar1, A H Nayfeh2 and S Ben Ayed1,2

1 Applied Mechanics and Systems Research Laboratory, Tunisia Polytechnic School, BP 743,

La Marsa 2078, University of Carthage, Tunisia2 Department of Engineering Science and Mechanics, MC 0219, Virginia Polytechnic Institute

and State University, Blacksburg, VA 24061, USA

E-mail: fehmi.najar@gmail.com

Received 4 June 2011, in final form 5 June 2011

Published 10 October 2011

Online at stacks.iop.org/SMS/20/115007

Abstract

Recently, piezoelectric cantilevered beams have received considerable attention for

vibration-to-electric energy conversion. Generally, researchers have investigated a classical

piezoelectric cantilever beam with or without a tip mass. In this paper, we propose the use of a

unimorph cantilever beam undergoing bending–torsion vibrations as a new piezoelectric energy

harvester. The proposed design consists of a single piezoelectric layer and a couple of

asymmetric tip masses; the latter convert part of the base excitation force into a torsion moment.

This structure can be tuned to be a broader band energy harvester by adjusting the first two

global natural frequencies to be relatively close to each other. We develop a

distributed-parameter model of the harvester by using the Euler-beam theory and Hamilton’s

principle, thereby obtaining the governing equations of motion and associated boundary

conditions. Then, we calculate the exact eigenvalues and associated mode shapes and validate

them with a finite element (FE) model. We use these mode shapes in a Galerkin procedure to

develop a reduced-order model of the harvester, which we use in turn to obtain closed-form

expressions for the displacement, twisting angle, voltage output, and harvested electrical power.

These expressions are used to conduct a parametric study for the dynamics of the system to

determine the appropriate set of geometric properties that maximizes the harvested electrical

power. The results show that, as the asymmetry is increased, the harvester’s performance

improves. We found a 30% increase in the harvested power with this design compared to the

case of beams undergoing bending only. We also show that the locations of the two masses can

be chosen to bring the lowest two global natural frequencies closer to each other, thereby

allowing the harvesting of electrical power from multi-frequency excitations.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Recently, there has been considerable interest in the field

of energy harvesting by converting ambient vibrations into

electrical energy. Williams and Yates [1] identified three

basic vibration-to-electric energy conversion mechanisms:

electromagnetic [2, 3], electrostatic [4], and piezoelectric [4, 5]

transduction. Mitcheson et al [3] indicated that piezoelectric

generators have a wider operating range at low frequencies

than other generators, especially when dealing with low-

frequency ambient vibrations. This type of transduction

does not require much space for implementation, thereby

making it suitable for a wide range of applications, such

as smart structures, electric resonators, filters, actuators, and

sensors [6]. Consequently, piezoelectric transduction has

received the most attention. For comprehensive reviews of its

literature, we refer the reader to Anton and Sodano [4], Beeby

et al [7], Sodano et al [8], Priya [9], Cook-Chennault et al [5],

and Erturk [10].

0964-1726/11/115007+11$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA1

Smart Mater. Struct. 20 (2011) 115007 A Abdelkefi et al

Many piezoelectric energy harvesters consist of a

cantilever beam with one piezoceramic layer (unimorph) or

two piezoceramic layers (bimorph). Although the piezoelectric

constitutive equations are nonlinear, most works in the

literature linearize these equations [11, 12]. Basically, the

beam is coupled to a vibrating host structure and the strain

induced in the piezoceramic layer(s) generates an alternating

voltage across the electrodes covering the piezoceramic

layer(s). Usually, the harvester is equipped with a proof mass,

which can be tuned so that the fundamental natural frequency

of the harvester beam is close to a dominant excitation

frequency available in the ambient vibration-energy spectrum.

Researchers have proposed various mathematical models

for beam energy harvesters [4, 7–10, 13]: lumped-parameter

and distributed-parameter models. The distributed-parameter

models are based on the Euler–Bernoulli theory for thin beams,

the Rayleigh formulation for slightly thicker beams, and the

Timoshenko theory for moderately thick beams. The Rayleigh

formulation accounts for rotary inertia and the Timoshenko

theory accounts for rotary inertia and transverse shear. The

distributed-parameter system is discretized by using either the

Rayleigh–Ritz method [14] or the Galerkin method [15, 16].

Both symmetric and asymmetric piezoelectric treatments

are considered and some studies account for longitudinal

displacements [10]. To the authors’ knowledge, none of the

existing studies consider harvesters that undergo bending–

torsion vibrations.

In this work, we design a unimorph cantilever beam-

based harvester that undergoes bending–torsion vibrations by

creating an offset between its center of gravity and the shear

center, thereby leading to a coupling between the bending and

torsion vibrations. Hence, there is a significant effect on the

natural frequencies, mode shapes, and response due to the

bending–torsion coupling. The offset is created by placing two

masses asymmetrically at the tip of the beam. We model the

beam–mass system by using the Euler–Bernoulli beam theory

and use Hamilton’s principle to derive the coupled governing

equations of motion and associated boundary conditions for

a base excitation. We calculate the exact mode shapes and

natural frequencies of the harvester and use these mode shapes

as basis functions in a Galerkin scheme to derive a reduced-

order model of the harvester. Closed-form expressions are

obtained for all needed outputs. Moreover, we demonstrate

that, when the lowest global frequencies are close to each

other, we have an interesting broadband frequency harvester.

Finally, we show that, when the asymmetry between the two

masses increases, the electrical harvested power as well as the

produced voltage increases.

2. System modeling

We consider a bi-layered cantilever beam in which one layer is

made of steel and the other layer is made of a piezoelectric

material. The piezoelectric layer is bonded to two in-plane

electrodes of negligible thickness. Two rigid masses are

asymmetrically placed at the tip of the beam. These masses

are interconnected using a rigid massless transverse beam, as

shown in figure 1. The clamped end of the beam is subjected

Figure 1. Schematic of the bending–torsion unimorph cantileverbeam.

Table 1. Physical and geometric properties of the unimorphcantilever beam.

Es Steel Young’s modulus (N m−2) 193 × 109

Ep Piezo Young’s modulus (N m−2) 66 × 109

Gs Steel Coulomb’s modulus (N m−2) 74.81 × 109

Gp Piezo Coulomb’s modulus (N m−2) 25.19 × 109

ρs Steel density (kg m−3) 8000

ρp Piezo density (kg m−3) 7800L Length of the beam (mm) 25Lc = a

2Tip mass offset (mm) 3.5

b Width of the beam (mm) 3.2hs Steel layer thickness (mm) 0.1hp Piezo layer thickness (mm) 0.4

e31 Piezoelectric coupling coefficient (C m−2) −12.54

∈s33 Piezoelectric permittivity (F m−1) 1.328 × 10−8

to a transverse harmonic displacement Y (t) = Y0 sin(�t). The

geometric and physical properties of the unimorph beam are

given in table 1.

The cantilever beam is flexible and undergoes coupled

bending–torsion motions. The tip masses and their attachments

are supposed to be rigid and asymmetric, thereby creating the

coupling between the bending and torsion motions. Next, we

use Hamilton’s principle

∫ t

0

(δT − δ�) dt = 0 (1)

to derive the governing equations of motion of the system and

their associated boundary conditions, where T is the system’s

kinetic energy and � is its potential energy. In the following,

the subscript s refers to the steel layer and the subscript p refers

to the piezoelectric layer.

We use three frame systems to describe the bending–

torsion motion of the beam with respect to its undeformed

configuration. The reference coordinate system (x, y, z) is

fixed to the clamped side of the beam, the second one

(x2, y1, z) is obtained by a rotation with respect to the z-

axis by the bending angle ψ , and the final coordinate system

(x2, y2, z2) is obtained by a rotation with respect to the x2-axis

by the torsion angle θ , as shown in figure 2.

2

Smart Mater. Struct. 20 (2011) 115007 A Abdelkefi et al

Figure 2. Rotation angles of the beam section.

We model the beam as an Euler–Bernoulli beam of length

L, width b, and height h, which comprises the thickness of

the steel layer hs and the thickness of the piezoelectric layer

hp. We let v(x, t) be the component of the local displacement

vector along the y-axis and assume relatively small rotation

angles so that ψ(x, t) = v′(x, t), where the prime denotes the

partial derivative with respect to x . For cylindrical bending and

torsion, we represent the local displacement vector associated

with an arbitrary point on the beam’s cross-section as

R = (Y + v)y + yy2 + zz2. (2)

The kinetic energy of the system is the sum of the kinetic

energies of the beam (T1) and the asymmetric masses (T2).

Variation of T1 is given by

δT1 = −∫ L

0

∫

A

ρ(y)RδR dA dx = −∫ L

0

(Avδv + Aθ1δθ) dx

(3)

where the over dot denotes the derivative with respect to time

t , ρ(y) is the beam density, A is the beam cross-section area,

Av = m(v + Y ), Aθ1= J1θ ,

J1 =2∑

i=1

∫ yi

yi−1

∫ 12

b

− 12

b

ρ(y)(y2 + z2) dy dz

and the beam’s rotary inertia is assumed to be negligible.

The asymmetric masses M1 and M2 are assumed to be

rigid and located at the distances d1 and d2 from the x-axis

(figure 1). They have a similar square cross-section area (a2)

in the (x2, y2) plane, but have different heights h1 and h2. Their

center of mass is located at x = L+Lc and their kinetic energy

is given as follows:

T2 = 12

M(vL + Y + Lcv′L )2 + 1

2I0θ

2L

+ Mt θL(vL + Y + Lcv′L) + 1

2I1v

′2L (4)

where the subscript L denotes variables calculated at x = L,

I1 = 112

(a2 + h21)M1 + 1

12(a2 + h2

2)M2, M = M1 + M2 is

the total tip mass, I0 = I1 + (M1d21 + M2d2

2 ) is the effective

polar second moment of area of the tip masses with respect

to the reference frame, and Mt = M1d1 − M2d2 describes

the asymmetry of the tip masses. Therefore, it follows from

equation (3) and variation of equation (4) that variation of the

total kinetic energy of the system is given by

δT = −∫ L

0

(Avδv + Aθ1δθ) dx

+ [M(vL + Y + Lcv′L ) + Mt θL ]δvL

+ [M Lc(vL + Y + Lcv′L ) + Mt LcθL + I1v

′L ]δv′

L

+ [I0θL + Mt (vL + Y + Lcv′L )]δθL . (5)

Variation of the potential energy of the whole system can

be expressed as

δ� =∫ L

0

∫

A

σ11δε11 dA dx

︸ ︷︷ ︸

δ�1

+∫ L

0

∫

A

σ12δγ12 + σ13δγ13 dA dx

︸ ︷︷ ︸

δ�2

(6)

where the strains and stresses in the steel and piezoelectric

layers are, respectively, given by

ε11 = zθv′′ − yv′′, γ12 = 2ε12 = −zθ ′,

and γ13 = 2ε13 = yθ ′,

σ s11 = Esε11, σ s

12 = 2Gsε12,

and σ s13 = 2Gsε13,

σp11 = Epε11 − e31 E2, σ

p12 = 2Gpε12,

and σp13 = 2Gpε13.

(7)

Here, Es and Ep are the Young’s moduli at constant electric

field, Gs and Gp are the shear moduli, and E2(t) = − V (t)

hpis

the electric field in the poling direction (V (t) is the potential of

the upper electrode and thus represents the voltage between the

piezoelectric’s electrodes). The electric displacement vector D

is related to the strains by

D =[

D1 = e15ε12

D2 = e31ε11+ ∈s33 E2

D3 = e15ε13

]

(8)

where the ei j are the piezoelectric stress coefficients and the

∈si j are the permittivities at constant strain.

Substituting equations (7) into (6) and integrating twice

the first integral by parts, we rewrite variation of the bending

potential energy δ�1 as

δ�1 =∫ L

0

[−M ′′b δv] dx + [M ′

bδv − Mbδv′]L

0 (9)

where Mb is the bending moment given by

Mb = −Ey3v′′ + 1

2e31b(y1 + y2)V [H (x) − H (x − L)],

Ey3= 1

3Esb(y3

1 − y30) + 1

3Epb(y3

2 − y31)

3

Smart Mater. Struct. 20 (2011) 115007 A Abdelkefi et al

Figure 3. Neutral axis position.

and H is the Heaviside step function. The positions of the

layers are defined in figure 3; they are given with respect to the

position of the neutral axis

y = (hp + hs)Ephp

2(Ephp + Eshs)+ hs

2

as follows:

y0 = −y, y1 = hs − y, y2 = (hs + hp) − y.

Variation of the torsion potential energy δ�2 is obtained

from equations (6) and (7) as

δ�2 = −∫ L

0

C ′δθ dx + [Cδθ ]L0 (10)

where C is the torsion moment given by

C = Ex2θ ′ = Gsθ

′[ 112

hsb3 + 1

3b(y3

1 − y30)]

+ Gpθ′[ 1

12hpb3 + 1

3b(y3

2 − y31)].

Substituting equations (5), (9), and (10) into (1) and

applying Hamilton’s principle, we obtain the following

equations of motion:

Ey3viv − 1

2e31b(y1 + y2)V (δ′(x)−δ′(x −L))+m(v+Y ) = 0,

(11)

J1θ − Ex2θ ′′ = 0 (12)

and their associated boundary conditions

v(0, t) = 0, v′(0, t) = 0, and

θ(0, t) = 0,(13)

Ey3v′′′

L − M(vL + Y + Lcv′L ) − Mt θL = 0, (14)

Ey3v′′

L + Lc M(vL + Y + Lcv′L ) + I1v

′L + Lc Mt θL = 0, (15)

Ex2θ ′

L + I0θL + Mt (vL + Y + Lcv′L ) = 0 (16)

where δ(x) is the Dirac function.

The two electrodes are connected to an external resistance

R. Hence, applying Gauss law yields

d

dt

∫

A

D ·n dA = −e31bhp

∫ L

0

v′′ dx− ∈s33

V bL

hp

= V

R(17)

where n is the vector normal to the plane (x2, z2).

For convenience, we introduce the following nondimen-

sional variables:

v = v

h, Y = Y

h, x → x

L, t → t

τ,

and τ = L2

√m

Ey3

into the equations of motion and boundary conditions, add

viscous damping terms, drop the hats, and obtain the following

nondimensional problem:

viv + c1v + v + Y − K V [δ′(x) − δ′(x − 1)] = 0, (18)

θ + c2θ − Ex2mL2

Ey3J1

θ ′′ = 0, (19)

v′′′1 − M

mL

(

v1 + Lc

Lv′

1 + Y

)

− Mt

hmLθ1 = 0, (20)

v′′1 + LcM

mL2

(

v1 + Lc

Lv′

1 + Y

)

+ I1

mL3v′

1 + LcMt

mhL2θ1 = 0, (21)

Ex2

Lθ ′

1 + I0

τ 2θ1 + Mt h

τ 2

(

v1 + Y + Lc

Lv′

1

)

= 0, (22)

− e31

bhhp

Lτ

∫ 1

0

v′′ dx− ∈s33

V bL

hpτ= V

R(23)

where K = 12e31b(y1 + y2)

L2

hEy3

and the subscript 1 indicates

variables calculated at x = 1.

3. Reduced-order model and analytical solution

To determine the electrical power produced by the proposed

design and to compare it with classical cantilever designs, we

solve the coupled structural–electrical equations (18), (19),

and (23) subject to the boundary conditions (20)–(22) for

a given base excitation Y (t). To accomplish this, we first

discretize the system using the Galerkin procedure. We have

at least two choices for the basis functions: the cantilever

mode shapes and the structural mode shapes that account for

the coupling between the bending and torsion motions caused

by the tip masses. Nayfeh et al [17] showed that, in such a

configuration, the cantilever mode shapes produce inaccurate

results for the asymmetric tip mass case. Therefore, we

employ the mode shapes that account for the bending–torsion

motions.

3.1. Structural natural frequencies and mode shapes

To investigate the dynamic response of the system, we start

by computing the structural natural frequencies and associated

mode shapes, taking into account the bending and torsion

coupling caused by the tip masses. To this end, we drop the

damping, forcing, and polarization from equations (18)–(23),

and let v(x, t) = φ(x)eiωt and θ(x, t) = ψ(x)eiωt . The

resulting eigenvalue problem is given by

φiv − ω2φ = 0, (24)

ψ ′′ + Ey3 J1

Ex2mL2

ω2ψ = 0 (25)

and associated boundary conditions

φ(0) = 0, φ′(0) = 0, ψ(0) = 0, (26)

φ′′′(1) + M

mLω2

(

φ(1) + Lc

Lφ′(1) + Mt

Mhψ(1)

)

= 0, (27)

4

Smart Mater. Struct. 20 (2011) 115007 A Abdelkefi et al

Figure 4. Variation of the natural frequencies of the structure with Mt : (a) first structural natural frequency, (b) second structural naturalfrequency, (c) third structural natural frequency, and (d) fourth structural natural frequency.

Table 2. The first four structural natural frequencies forMt = −0.1 g.m.

ω1 ω2 ω3 ω4

0.245 332 1.099 67 3.235 71 22.8913

φ′′(1) − ω2

mL2

×(

M Lcφ(1) − (L2c M + I1)

Lφ′(1) − Mt Lc

hψ(1)

)

= 0,

(28)

ψ ′(1) − I0 Lω2

Ex2τ 2

ψ(1) − hL Mt

Ex2τ 2

ω2

(

φ(1) + Lc

Lφ′(1)

)

= 0.

(29)

The general solutions of equations (24) and (25) can be

expressed as

φ(x) = a1 sin(√

ωx) + a2 cos(√

ωx)

+ a3 cosh(√

ωx) + a4 sinh(√

ωx) (30)

ψ(x) = a5 cos(βx) + a6 sin(βx), (31)

where the ai are arbitrary constants and β2 = E y3 J1ω2

Ex2mL2 .

Substituting equations (30) and (31) into (26)–(29) and setting

the determinant of the resulting coefficient matrix equal to

zero, we obtain the characteristic algebraic equation governing

the eigenvalues ω. Solving numerically this characteristic

equation yields the natural frequencies of the structural part

of the harvester. Using these frequencies, we calculate the

structural modes φ(x) and ψ(x). In table 2, we list the first

four structural natural frequencies when Mt = −0.1 g.m.

Next, we investigate the effect of Mt on the structural

natural frequencies. To this end, we use the parameters listed

in table 1, let d2 = 25 mm, M1 = M2 = 5 g, and plot

variation of the resulting natural frequencies with Mt obtained

Table 3. Analytical and FE comparison of the first two structuralnondimensional natural frequencies.

d1 (mm) ω1 ω1 (FE) ω2 ω2 (FE)

10 0.388 998 0.391 644 2.322 38 2.304 1620 0.386 764 0.403 766 1.530 86 1.538 40

by varying d1. The heights h1 and h2 of the tip masses

are calculated for the chosen masses M1 and M2 and lead

density (11 350 kg m−3). We recall that the asymmetry of

the tip masses can be realized by either letting M1 �= M2

or d1 �= d2 (which is our case). In figure 4, we show

variation of the lowest four structural frequencies with Mt , and

in figure 5, we show the modes shapes obtained for Mt =−0.1 g.m.

3.2. FE modal validation

A numerical method is used to validate the analytically

obtained structural natural frequencies and mode shapes for

different bending–torsion configurations. For this purpose, an

FE model is developed using the commercial software Ansys.

A 3D model is built with a tetrahedral mesh. Figure 6 shows

the FE model of the harvester when d1 = 20 mm, d2 = 8 mm,

and M1 = M2 = 3 g. Figure 7 detail the first two mode shapes

of the structure. A good agreement is shown in table 3 between

the analytical and FE results for two distinct cases obtained

by changing only the value of the distance d1 (d1 = 10 and

20 mm) when d2 = 8 mm and M1 = M2 = 3 g.

3.3. Discretized equations

To develop a reduced-order model of the system, we express

the displacement v(x, t) and twist angle θ(x, t) using the

Galerkin procedure in the form

5

Smart Mater. Struct. 20 (2011) 115007 A Abdelkefi et al

Figure 5. The first four mode shapes of the structure.

Figure 6. FE model of the bending–torsion unimorph cantileverbeam when d1 = 20 mm, d2 = 8 mm and M1 = M2 = 3 g.

v(x, t) =2∑

i=1

φi(x)qi(t) and

θ(x, t) =2∑

i=1

ψi (x)qi(t)

(32)

where the test functions are the structural mode shapes

computed by the procedure in section 3.2. The choice

of two structural mode shapes to discretize the system is

related to the choice of applied excitation frequencies, which

are, in general, low compared to the third and higher

structural natural frequencies. To reduce the algebra, we

perform the discretization by substituting equation (32) into

the nondimensional Lagrangian ℓ = T − � of the system,

carrying out the spatial integration, and then writing down the

Euler–Lagrange equations

d

dt

∂ℓ

∂ qi

− ∂ℓ

∂qi

= 0, i = 1, 2. (33)

Then, we add a linear damping term in each equation. The

damping coefficient is determined using the relation ci = ωi

Q,

where Q = 50 is the quality factor. The resulting discretized

equations can be expressed as

Ai qi + ci qi + Bi qi − Ki V + Fi Y = 0, i = 1, 2 (34)

where

Ai =∫ 1

0

φi (x)2 dx + J1

mh2

∫ 1

0

ψi (x)2 dx + I0

mLh2ψi (1)2

+ I1

mL3φ′

i (1)2 + M

mL

(

φi(1) + Lc

Lφ′

i(1)

)2

+ 2Mt

mLhψi (1)

(

φi (1) + Lc

Lφ′

i (1)

)

,

Bi =∫ 1

0

φ′′i (x)

2dx + Ex2

L2

Ey3h2

∫ 1

0

ψ ′i (x)

2dx,

Ki = Kφ′i (1),

Fi =∫ 1

0

φi (x) dx + M

mL

(

φi(1) + Lc

Lφ′

i(1)

)

+ Mt

mLhψi (1).

Discretizing the Gauss law, equation (23), yields

e31

bhph

τ L[q1φ

′1(1) + q2φ

′2(1)]+ ∈s

33

V bL

hpτ+ V

R= 0. (35)

3.4. Analytical solution

We use the discretized equations to determine closed-form

expressions for the displacement, twist angle, voltage, and

harvested power for a harmonic base excitation Y (t) =Y0 sin(�t), where � is the excitation frequency and Y0 is its

nondimensional amplitude. We note that Y = Im(Y0ei�t ),

where Im stands for the imaginary part. Consequently, we

seek the solution of equations (34) and (35) in the form q1 =Im(Q1ei�t ), q2 = Im(Q2ei�t ), and V = Im(V0ei�t ) and obtain

the following system of algebraic equations:[

B1 − A1�2 + i�c1 0 −K1

0 B2 − A2�2 + i�c2 −K2

A3i� A4i� 1R

+ i�B3

]

×[

Q1

Q2

V0

]

= �2Y0

[F1

F2

0

]

(36)

6

Smart Mater. Struct. 20 (2011) 115007 A Abdelkefi et al

Figure 7. The first two mode shapes of the energy harvester obtained with FE when d1 = 20 mm, d2 = 8 mm, and M1 = M2 = 3 g. (a) Leftview of the first mode shape. (b) Front view of the first mode shape. (c) Left view of the second mode shape. (d) Front view of the secondmode shape.

Figure 8. Frequency–response curve of the harvested power near thefirst natural frequency for different values of the load resistance.

where

A3 = e31

hbhp

τ Lφ′

1(1), A4 = e31

hbhp

τ Lφ′

2(1),

B3 = ∈s33

bL

hpτ.

4. Results: symmetric case

For the symmetric case, the harvester behaves as a classic

unimorph piezoelectric harvester with a tip mass. We note

that the effect of the torsion is not observable because only

the bending modes are excited since Mt = 0 and Fi does not

Figure 9. Frequency–response curve of the harvested power near thesecond natural frequency for different values of the load resistance.

depend on the torsion mode shape ψ . However, the symmetric

case serves as a reference when investigating the performance

of the new asymmetric design.

We use the set of parameters listed in table 1 with M1 =M2 = 3 g and d1 = d2 = 5 mm. We show in figures 8

and 9 zooms near the first and second natural frequencies of the

variation of the normalized harvested electrical power with the

excitation frequency for different values of the load resistance.

As expected, an increase in the load resistance results in a

change in the harvested power. The normalized maximum

electric harvested power Pmax is calculated as

Pmax = V 2maxτ

4

RhY 20 �4

7

Smart Mater. Struct. 20 (2011) 115007 A Abdelkefi et al

Figure 10. Variation of the normalized maximum harvestedelectrical power with the load resistance for the open-circuit andshort-circuit global frequencies.

Table 4. Global natural frequencies for open- and short-circuitconfigurations when M1 = M2 = 3 g and d1 = d2 = 5 mm.

ω1 ω2

Open-circuit frequency (R = 107 �) 0.344 81 5.148 75Short-circuit frequency (R = 102 �) 0.321 53 5.071 30

where Vmax is the maximum value of the harvested voltage at

steady state.

The global natural frequencies (obtained from the

frequency–response curves) of the harvester in open- and

short-circuit configurations corresponding to the resonance

frequency of the system when R = 107 and 102 � are listed in

table 4. We note that the maximum harvested electrical power

that could be generated is obtained for R = 1420 �, which

corresponds to Pmax = 0.135 61 W s4 m−2 at � = 0.321 53

(short-circuit frequency). However, this configuration does

not correspond necessarily to the best configuration because,

at this resistance load, the harvested power has a sharp peak

at the resonance frequency and drops drastically for all other

excitation frequencies, as observed in figures 8 and 9. For

practical applications, because the excitation is, in general, not

harmonic, another resistance might be more suitable to harvest

the maximum power.

Next, we investigate the dependence of the maximum

harvested electrical power Pmax on the load resistance R; we

vary it from open-circuit to short-circuit values. In figure 10,

we show variation of Pmax with R for two fixed excitation

frequencies, namely those corresponding to the open- and

short-circuit configurations given in table 4.

5. Results: asymmetric case

In this section, we investigate the influence of the asymmetry

Mt of the tip masses on the dynamic behavior of the harvester

Figure 11. Variation of the first four structural natural frequencieswith Mt .

and thus on the harvested electrical power. The results of

the asymmetric case are compared with those obtained in the

symmetric case. We vary the asymmetry parameter Mt of the

tip masses by keeping M1 = M2 = 3 g and d2 = 5 mm and

only vary d1.

We start by showing, in figure 11, variation of the first

four structural natural frequencies of the system with the

tip asymmetry parameter Mt , which is varied by varying d1

from −50 to 50 mm. We note here that negative values of

d1 correspond to placing both masses on the same side of

the cantilever beam. When Mt = 0, the structural natural

frequencies are ω1 = 0.321 53, ω2 = 3.870, ω3 = 5.071 28,

ω4 = 23.6480. When Mt is increased positively, the second

frequency is greatly affected because it is dominated by the

torsion motion, as seen in the mode shapes in figure 5. From

figure 11, we also observe three different regions. Starting

from d1 = 50 mm, first, the curve of the second natural

frequency comes very close to that of the third frequency,

but without crossing it; that is, the two curves veer from one

another. Then, the second natural frequency asymptotes a

constant value and the third natural frequency increases. The

third region corresponds to a reduction in the second natural

frequency only. The first and fourth natural frequencies seem

to be slightly affected by variations in Mt . We note that the

third natural frequency is not affected by positive variations

in d1 but greatly affected by small negative values of d1. To

validate the proposed analytical dynamic solution, we compare

the frequency–response curves of the asymmetric system when

d1 = 20 mm, d2 = 8 mm and M1 = M2 = 3 g. The

corresponding natural frequencies obtained using the analytical

and FE approaches are shown in table 3. In figure 12,

we show the variation of the maximum nondimensional tip

deflection as a function of the normalized excitation frequency.

The normalization of the excitation frequency is obtained by

dividing � by the corresponding natural frequency to the

calculation method chosen from table 3. We note that there

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Smart Mater. Struct. 20 (2011) 115007 A Abdelkefi et al

Figure 12. Variation of the maximum nondimensional tip deflection when d1 = 20 mm, d2 = 8 mm and M1 = M2 = 3 g near (a) the firstpeak and (b) the second peak for two different approaches: analytical prediction (continuous lines) and FE Ansys model (red points).

Figure 13. Frequency–response curve of the maximumnondimensional tip deflection for different values of d1 whenR = 104 �.

is a good agreement between both methods for excitation

frequency values close to resonance.

Next, we study the dynamic behavior of the asymmetric

system for different values of d1 ranging from 10 to 50 mm

when the electrical load resistance is equal to 104 �. In

figures 13–16, we show, respectively, variations of the

nondimensional tip deflection, nondimensional tip rotation,

normalized harvested voltage, and Pmax with the excitation

frequency �. In these figures, we note the appearance of a

second peak, in the frequency–response curves, corresponding

to the global natural frequency of the second mode, which is

dominated by torsion. Variation of d1 is accompanied with

a shift in the global natural frequencies, as seen in figure 11.

The two peaks are separated by an antiresonance for the cases

of the tip deflection, the voltage, and the harvested electrical

power. On the other hand, the antiresonance occurs outside the

band of the two peaks in the case of the tip rotation except for

d1 = 10 mm where it is inside the band.

We note that increasing d1 results in (a) an increase

in the maxima of all of the system responses (i.e., tip

Figure 14. Frequency–response curve of the maximum tip rotationfor different values of d1 when R = 104 �.

displacement, tip rotation, voltage, and harvested power) and

(b) a decrease in the difference between the lowest two

global natural frequencies and hence the frequency difference

between the two peaks in the responses, as seen in figures 13–

16. Consequently, the asymmetry not only increases the

harvested power from a single-frequency environment, but also

from a multi-frequency environment.

In figure 17, we show the effect of the load resistance

on the normalized maximum harvested electrical power by an

asymmetric harvester for a fixed excitation frequency when

M1 = M2 = 3 g, d1 = 45 mm, and d2 = 5 mm. We chose

two frequencies, namely those corresponding to the open- and

short-circuit global natural frequencies shown in table 5. We

observe results similar to those found for the symmetric case

for both frequencies.

To compare the efficiency of the proposed design with tip

mass asymmetry with that of the symmetric case, we plot in

figure 18 variations of the maximum normalized maximum

harvested electrical power Pmax at resonance with the tip mass

asymmetry parameter Mt . For the harvested power at the first

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Smart Mater. Struct. 20 (2011) 115007 A Abdelkefi et al

Figure 15. Frequency–response curve of the maximum normalizedharvested voltage for different values of d1 when R = 104 �.

Figure 16. Frequency–response curve of the normalized maximumharvested electrical power for different values of d1 whenR = 104 �.

Table 5. Natural frequencies for open- and short-circuitconfigurations when M1 = M2 = 3 g, d1 = 45 mm and d2 = 5 mm.

ω1 ω2

Open-circuit frequency 0.327 06 0.906 29Short-circuit frequency 0.306 86 0.898 82

natural frequency, the benefit of the tip mass asymmetry is very

clear because the minimum value of Pmax is obtained for the

symmetric case (Mt = 0). For the case when d1 = −50 mm

(Mt = −0.165 g.m), an increase of 30% in Pmax is observed.

For the harvested power at the second natural frequency, a

more complex behavior is observed. In fact, at d1 = −20 mm

Figure 17. Variation of the normalized maximum harvestedelectrical power with the load resistance for the open-circuit andshort-circuit global natural frequencies.

(Mt = −0.075 g.m) and d1 = 10 mm (Mt = 0.015 g.m), we

observe a drastic drop in Pmax. In these configurations, the first

mass M1 is located symmetrically with respect to the second

mass M2. For these specific values, the antiresonance is located

on top of the second natural frequency and thus destroys the

contribution of the torsion motion.

6. Conclusion

We propose the use of a unimorph piezoelectric cantilever

beam undergoing bending–torsion vibrations induced by

asymmetric tip masses for energy harvester application.

Using the Euler–Bernoulli beam theory and Hamilton’s

principle, we derive the governing coupled equations of motion

and associated boundary conditions under harmonic base

excitations. We compute the structural natural frequencies

and associated mode shapes accounting for bending–torsion

coupling and validate them using the FE software Ansys.

These mode shapes are used as basis functions in a Galerkin

procedure to obtain a reduced-order model of the coupled

structural–electrical system, which provides the global natural

frequencies and mode shapes of the harvester. Then, closed-

form expressions are obtained for the displacement, twisting

angle, voltage output, and harvested electrical power. The

analytical expressions are used to demonstrate enhancement

of the harvester by the asymmetry of the mass distribution.

A parametric study is performed to ascertain the influence of

the load resistance and asymmetry on the tip displacement,

twisting angle, voltage, and harvested power. We found that the

improvement in the harvester’s performance increases as the

asymmetry increases. In fact, we found mass distributions that

increase the harvested power by as much as 30%. Moreover,

we found that the lowest two global natural frequencies move

closer to each other as the asymmetry increases, implying the

ability of the harvester to harvest power from a broader range

of excitation frequencies.

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Smart Mater. Struct. 20 (2011) 115007 A Abdelkefi et al

Figure 18. Variation of the normalized maximum harvested electrical power at resonance with the asymmetry for the first (a) and second (b)global natural frequencies when R = 104 �.

Acknowledgment

The authors would like to thank Prof. Mohammad R Hajj

(Department of Engineering Science and Mechanics, Virginia

Tech) for his assistance concerning the ANSYS simulations.

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