Smart Materials and Structures Volume 21 Issue 6 2012 [Doi 10.1088_0964-1726_21!6!065017] Xu, Jia...

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Optimization of a right-angle piezoelectric cantilever using auxiliary beams with different

stiffness levels for vibration energy harvesting

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IOP PUBLISHING SMARTMATERIALS ANDSTRUCTURES

Smart Mater. Struct. 21 (2012) 065017 (13pp) doi:10.1088/0964-1726/21/6/065017

Optimization of a right-angle

piezoelectric cantilever using auxiliarybeams with different stiffness levels forvibration energy harvesting

Jia Wen Xu, Yong Bing Liu, Wei Wei Shao and Zhihua Feng

Department of Precision Machinery and Precision Instruments, University of Science and Technology of

China, 230026, Anhui, Peoples Republic of China

E-mail: [email protected]

Received 7 July 2011, in final form 16 January 2012

Published 22 May 2012

Online atstacks.iop.org/SMS/21/065017

Abstract

This paper presents experiments and models of a piezoelectric cantilever generator with a

right-angle structure. Analysis shows that the extended part provides a large torque to the main

beam, which can dramatically smoothen the strain distribution of the main beam. The

auxiliary beam was fabricated with half the length of the main beam. When the auxiliary beam

has a stiffness which is 0.02 times that of the main beam the piezoelectric element has a highly

uniform strain distribution; in addition, its relative utilization efficiency (RUE) is 93% at the

initial resonant frequency, whereas it is 50% for a conventional rectangular piezoelectriccantilever. The performances of three right-angle generators with auxiliary beams having

different levels of stiffness, but constant-stiffness main beams are studied. The RUE of the

piezoelements increases as the auxiliary beams stiffness decreases. A model based on the

RayleighRitz method is established to demonstrate the principle of the strain-smoothing

effect. The voltage and power outputs of the generators are measured. Finite element method

simulations are also presented, and the result fits the experiments well.

(Some figures may appear in colour only in the online journal)

1. Introduction

Wireless apparatuses and portable electronic devices are

mainly powered by conventional batteries. However, replac-

ing or recharging batteries in some situations can become

tedious and expensive. Moreover, batteries offer a limited

energy supply, which means a limited duration for the duty

cycle of the device. Thus, much effort is necessary to seek

a simple way to solve the problem. Recent developments

in low-power electrics that allow the operation of these

devices with much less power have made the application of

microgenerators possible.There exists usable kinetic energy within industrial

machines, human activities and environmental vibration. Thedevelopment of on-site generators has been emphasized

because they can transform available kinetic energy intoelectrical energy [13]. Lead zirconate titanate (PZT) energy

harvester has emerged as one of the primary methods for

harvesting energy. This is superior to other means of energy

harvesting, such as electromagnetic and electrostatic, due

to its simple structure, high electromechanical coupling

coefficient, non-reliance on extra voltage sources and

easy integration into a system [46]. In PZT-based

energy harvesting, different mechanical structures have

been presented to transform vibration energy into electric

energy [715]. Among these structures, a cantilever beam

with a piezoelectric element is a common configuration

because it has a relatively low working frequency and highaverage strain.

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Smart Mater. Struct. 21 (2012) 065017 J W Xuet al

However, the conventional rectangular cantilever has

many disadvantages due to its inherent attributes. The

piezoelectric element could not be fully utilized. Efficiency

is considerably low because its longitudinal strain distribution

changes linearly from the clamped end to the free end; and

it is zero at the free end [2]. With the aim of obtaining the

highest output power per unit of PZT element, many novelstructures have been proposed, and they are usually optimized

in the shapes of the cantilever. Roundy et al discussed

one optimal structure with uniform strain distribution, in

which a trapezoidal structure was considered. For every

unit volume of PZT, the trapezoidal cantilever can supply

twice the energy output of a rectangular structure [16].

Frank further analyzed and tested this concept. He proved

that, compared with the conventional cantilever, 30% more

power can be obtained from the trapezoidal cantilever at a

resonant frequency of under 2.5 g amplitude by a vibration

source [17]. Wood et al, Benasciutti et al and Dietland et alalso discussed the advantages of a trapezoidal cantilever over

a rectangular one [1820]. A method using a tapered beamwas introduced by Mehraeen [21]. He proved that this type

of device could also achieve a uniform longitudinal strain

distribution in the piezoelectric element. Zhao also proposed

a type of stair-shaped beam that had twice the efficiency

of a conventional cantilever in a quasi-stationary state [22].

Concurrently, Paquin and Friswell analyzed the performance

of the tapered beam model [23,24].

A conventional rectangular cantilever would also have

uniform strain distribution if it was excited by a torque

applied at its free end in the quasi-stationary state. In the

current study, a right-angle cantilever is reported, and this

device has an extended part forming a right angle with themain beam. At the series resonant frequency of the main

beam and the extended part, the main beam works in a

quasi-stationary state. Moreover, the auxiliary beam could

provide a considerably large shaking moment directly to the

main beam. Thus, a highly uniform strain distribution of the

main beam could be obtained and the performance of the

energy harvester could be considerably improved.This paper proceeds as follows. First, a simple model

based on the RayleighRitz method for the piezoelectric

compound structure is proposed and used to demonstrate

the principle of the strain-smoothing effect. Next, finite

element method (FEM) simulation is presented to examine the

strain distributions of three devices at their respective initialresonant frequencies. The three devices are made of auxiliary

beams of different stiffness and main beams with constant

stiffness. Experiments are carried out to study the strain

distributions and relative utilization efficiency (RUE) of the

three devices. Finally, their voltage output and power output

under sinusoidal acceleration of 2.5 m s2 are examined and

compared.

2. Prototype structure

Figure1 is a schematic diagram of the right-angle cantilever.

As can be seen, the device has an auxiliary beam in the

orthogonal direction of the main beam, which is regardedas a cantilever fixed to a basement. The auxiliary beam has

Figure 1. Schematic diagram of right-angle cantilever.

one end fixed to the free end of the main beam and the

other end installed with a proof mass. The main beam and

the piezoelectric layer are geometrically uniform in their

longitudinal directions, and the poling axis of the piezoelectric

element is perpendicular to the main beam.

The sinusoidal base movement of constant amplitude

is orthogonally decomposed into Uxx-direction vibration

and Uzz-direction vibration (figure 1). The x-direction is

perpendicular to the main beam, whereas the z-direction is

parallel to it. In this case, Ux and Uz have the same phases,

but they have different amplitudes. They obey the following

conditions:

Ux(t)= U(t) cos, (1)

Uz(t)= U(t) sin. (2)

3. System modeling

3.1. Mathematical model

Models of piezoelectric cantilevers based on the Euler

Bernoulli beam theory have been studied by many

researchers [2530], and these distributed parameter models

demonstrate great accuracy. One example is the L-shaped

piezoelectric cantilever model presented by Erturk and

Inman[31]. In this section, a mathematical model based on

the EulerBernoulli beam theory is also established to analyze

the right-angle cantilever proposed in section2.

The following study is based on the assumptions: (1) theprototype is considered as a composite EulerBernoulli beam;

therefore, the effects of shear deformation and rotary inertia of

the beams are neglected; (2) the PZT layer is perfectly bonded

to the substructure layer; thus, there is no shear strain between

the layers; and (3) no deformation is presented at the joint of

the main beam and the auxiliary beam.

In figure2(a),l1, bm, l2, ba,Mp, hp, hb, ha, p, b andarepresent the following parameters, respectively: length and

width of the main beam, length and width of the auxiliary

beam, proof mass, height of the PZT and titanium substructure

of the main beam, height of the auxiliary beam, mass density

of the substructure, the PZT plate and the auxiliary beam.

The cross-section of the composite beam is shown infigure 3. The coordinate system XOZ is set up with the

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Figure 2. (a) Schematic drawing of the right-angle cantilever, (b) equivalent model of the main beam and the auxiliary beam.

Figure 3. (a) Cross-section of the main beam for the determinationof the neutral plane position. (b) Cross-section of the auxiliary

beam.

origin at the neutral axis NN, and the distance between

the bonding layer and the neutral axis is a. Here, a is

determined by calculating the stress in the cross-section of the

unimorph [28]. Thus,

a=1

2

Ebh2b Eph

2p

Ebhb+ Ephp. (3)

Considering the composite layer of the PZT and the base beam

as one beam with flexural rigidityEIeq = 13 bEb(h

3b 3ah

2b+

3a2hb)+13 bEp(h3p + 3ah2p + 3a2hp),EIarepresents the rigidityof the auxiliary beam given byEIa =

112 baEah

3a .

w1(zm, t) and w2 = (xa, t) denote the transverse

deflections of the main beam and the auxiliary beam,

respectively, at locations x,z and time t in the coordinate

systemXOZlocated at the base. Equations of motion for the

system are derived by extended Hamiltons principles. In this

model, the main beam is considered as a fixed-free cantilever;

the auxiliary beam is considered as a hinged-free cantilever,

as shown in figure2(b).

The kinetic and potential energy of the system under base

excitations ofUxand Uz are respectively expressed as

T =1

2

l1

0m1

w1(zm, t)

t+

Ux

t

2

dzm

+1

2

l10

m1

Uz

t

2dzm

+1

2

l20

m2

Uz

t+

w2(xa, t)

t

2dxa

+1

2Mp

Uz

t+

w2(l2, t)

t

2

+1

2(m2l2+ Mp)

w1(zm, t)

t+ Ux

2

, (4)

V =1

2

l10

EIeq

2w1(zm, t)

z2m

2dzm

+1

2

l20

EIa

2w2(xa, t)

x2a

2dxa, (5)

wherem1and m2are the mass densities per unit length for the

composite beam, given respectively as

m1 = (php+ bhb)bm,

m2 = abaha.

If the viscous damping coefficient for the main beam isdenoted as c1 and the viscous damping coefficient for the

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auxiliary beam is denoted as c2, then the virtual work is

Wc =

l10

c1w1(zm, t)

tw dzm

l20

c2w2(xa, t)

tw dxa, (6)

where w is the virtual displacement. Using extendedHamiltons principles, we arrive at t2

t1

(T V+ Wc) dt= 0. (7)

Thus, the motion equation for the vibration of the main beamand the auxiliary beam, including the effect of damping, canbe written respectively as

2w21(zm, t)

z2m+c1

w1(zm, t)

t+EIeq

4w1(zm, t)

z4m=0,

for 0 zm l1, (8)

2w22(xa, t)

x2a+c2

w2(xa, t)

t+EIa

4w2(xa, t)

x4a=0,

for 0 xa l2. (9)

Modes of the undamped free vibration are typically usedto estimate the modes of the devices at resonance frequencies.In the case of undamped free vibration, Uz = Ux = 0, andthere is no displacement or rotation at the clamped end of themain beam relative to the base; thus, the boundary conditionsare as follows:

w1(0, t)=w1(zm, t)

zm

zm=0

=0, (10)

w2(0, t)= 0,w1(zm, t)

zm

zm=l1

=w2(xa,t)

xa

xa=0

, (11)

(m2l2+Mp)2w1(zm, t)

t2

zm=l1

EIeq3w1(zm, t)

z3m

zm=l1

=0,

(12)

EIeq2w1(zm, t)

z2m

zm=l1

= EIa2w2(xa, t)

x2a

xa=0

, (13)

EIa3w2(xa, t)

x3a

xa=l2

Mp2w2(xa, t)

t2

xa=l2

=0, (14)

EIa2w2(xa, t)

x2a

xa=l2

=0. (15)

The vibration response relative to the base of the levercan be represented as an absolutely convergent series of theeigenfunctionsmr(zm)andar(xz). Thus,

w1(zm, t)=

r=1

mr(zm)rx(t), (16)

w2(xa, t)=

r=1

ar(xa)rz(t), (17)

where mr(zm) and ar(xz) denote the mass normalizedeigenfunction of the rth vibration mode of the main beam

and the auxiliary beam, respectively. Here, the influence of

gravity on the shapes of the beams is ignored, because gravity

contributes little to the sinusoidal voltage output; r(t) is the

modal mechanical response expression of the device of the

rth vibration mode, and it is the same for both sections of the

device. Eigenfunctionsmr(zm)andar(xz)for the main beam

and the auxiliary beam are denoted respectively asmr(zm)= Amrsin rzm+ Bmrcos rzm

+ Cmrsinhrzm+ Dmrcoshrzm, (18)

ar(xa)= Aarsin rxa+ Barcos rxa

+ Carsinhrxa+Darcoshrxa, (19)

where 4r = 2r

m1EIeq

, 4r = 2r

m2EIa

and r is the undamped

resonance frequency of the rth vibration mode in open-circuit

conditions. In addition, Amr, Bmr, Cmr, Dmr, Aar, Bar, CarandDarshould be evaluated by the aforementioned boundary

conditions. Thus,

Amr= Cmr, Bmr= Dmr, Bar= Dar.

Eigenfunctionsmr(zm)andar(xa)can be simplified as

mr(zm)= Bmr(cosrzmcosh rzm)

+Amr(sinrzmsinhrzm), for 0 zm l1, (20)

ar(xa)= Aarsin rxa+ Barcos rxa+ Carsinhrxa

Barcosh rxa, for 0 xa l2. (21)

The eigenfunctions can be further simplified. The Maclaurin

series expansions of the two expressions are O(z3) of the

second term. The derivation of the first two terms of the

expansion from cosrzm, cosh rzm, sin rzm and sinhrzm

is the general solution of the undamped system, which fits thegeneral boundary condition. Thus, the eigenfunctionsmr(zm)

at the first resonance frequency can be written as

m1(zm)= A1z3m+B1z

2m+ C1zm+ D1,

for 0 zm l1. (22)

In addition,ar(xa)also has a similar simplified form

a1(xa)= A2x3a + B2x

2a + C2xa+ D2,

for 0 xa l2. (23)

Hence,C1 = D1 = D2 =0, and

B2 = 3A2l2,(m2l2+ Mp)(A1l

31+ B1l

21)EIaA2

=Mp(A2l32+B2l

22+ C2l2)EIeqA1,

C2 =3A1l21+2B1l1,

EIaB2 =EIeq(3A1l1+ B1).

The eigenfunctions have one degree freedom, which can

be evaluated by normalizing them according to the following

orthogonality conditions: l10

m1mi(zm)mj(zm) dzm+

l20

m1ai(ya)aj(ya) dya

+ (Mp+ m2l2)mi(l1)mj(l1)+Mpai(l2)aj(l2)=i,j, (24)

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Figure 4. Theoretical diagram ofk= EIaEIeq

versusar.

and l10

EIeq

mi(zm)

mj(zm) dzm

+

l20

EIa

ai(ya)

aj(ya) dyaEIami(l1)

aj(0)

+EIeqmi(0)

aj(0)= i,j

2r. (25)

Here, i,j is the Kronecker delta defined as unity when i = jand zero wheni =j.

The PZT element is bonded to the main beam and works

under the initial resonance frequency; thus, the eigenfunction

m1(zm) should be further studied. This expression can be

described as:

m1(zm)= B1(z2m+ arz

3m), ar=

A1

B1,

for 0 zm l1. (26)

The mathematical expression of the stress distribution of themain beam is:

Sm1(zm)= h2m1(zm)

z2m=2hB1(1+3arzm),

for 0 zm l1. (27)

From the expression, the coefficient arevidently determines

how the strain distribution has been smoothed at the firstresonance frequency. Notably, the expression of the strain

distribution for a conventional cantilever (with the same

parameters as the main beam in this experiment) is also S(x)=

2hB1(1 + 3arx), where a

r =

13l1

. Using the parameters

presented in section5.1, the a rfor a conventional cantilever

is 11.1. The strain distribution is a function of the ratio of

EIa andEIeq, a smaller |ar|indicates a more smoothed stress

distribution. The relation betweenk= EIaEIeq

andaris presented

in figure4.Figure 4shows that a smaller kdenotes a smaller |ar|,

thereby indicating a more smoothed strain distribution. The

physical meaning of this strain-smoothing effect will beexplained in section3.2.Remarkably, this expression of strain

distribution only applies at the first resonant frequency of

the device, with the assumption that it is in a low force

environment. In addition, the strain distribution may change

at other frequencies, as Erturk has described in a previous

study [29].

3.2. Force and torque applied to the main beam

The study of the transient dynamic characteristic of a PZT

cantilever utilizing the impedance method has been performed

in previous studies, and the proposed models have shown

fairly good accuracy [19, 32, 33]. The impedance method

has been studied and implemented in the present research

to get the corresponding displacement of the device under

a generalized force, as well as the interaction between the

main beam and the auxiliary beam. The forcedisplacement

impedance is used to study the models because the electric

charge generated is proportional to the amplitude of the

strain of the device. The forcedisplacement impedance and

admittance are defined as follows [34]:

Z(j)=F(j)

X(j), (28)

H(j)=1

Z(j). (29)

The model in section 2.1 analyzes the dynamic response

of the device in the following discussions. When the device

is solely excited by a vibration in the z-direction, the base

movement first excites the auxiliary beam, which then drives

the main beam. When the device is solely excited by

x-direction vibration, the excitation of the auxiliary beamdepends on its own inertial force. However, the auxiliary

beam would also provide the main beam with a large torque.

Considering the energy dissipation in the beammass system,

a viscous damping factor is added. The modal damping ratio

is identified directly from the frequency response or time

domain measurement.

Figure 5(a) shows the model of impedance of the

right-angle cantilever; figure5(b) is the equivalent circuit of

the impedance of the device [34]. Here, [M1] is a matrix of

the generalized mass of the main beam, [K1] is a matrix of

the generalized stiffness coefficient of the main beam, [c1]is

a matrix of the generalized damping coefficient of the mainbeam,[Mp] is a matrix of the generalized mass of the proof

mass,[M2]is a matrix of the generalized mass of the auxiliary

beam,[K2] is a matrix of generalized stiffness coefficient of

the auxiliary beam and [c2] is a matrix of the generalized

damping coefficient of the auxiliary beam in this device.

[c2] can be ignored in comparison to [c1]. The mass of the

auxiliary beam is much smaller than the proof mass, thus, it

is also ignored in the analysis. {F} is the generalized force

applied at the device. There exist both force and torque at

the joint point of the main and auxiliary beams; hence, the

generalized force is defined as follows:

{F} =

FxFz

= Mp

Ux(t)Uz(t)

. (30)

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Figure 5. (a) Schematic model of the impedance of the right-angle cantilever, (b) equivalent circuit of impedance of the device.

The response of the device {} =xz

obeys the

following equation:[Z0] {} = {F}, where[Z0] = 2[Mp]+

[Z1][Z2][Z1]+[Z2]

[34].

The displacement impedance of the main beam and

auxiliary beam can be calculated using the following

principles[34]:

[Z1] = [K1] +j[c1] 2[M1], (31)

[Z2] = [K2] +j[c2] 2[M2]. (32)

To calculate the generalized stiffness coefficients and

generalized mass of the device, the following model is used:

[K1] =

l10

EIeq(m(z))

2 dz

l10

EIeq(m(z))

2

m(z)dz

l10

EIeq

(m(z))2

m(z)d l1

0EI

eqm(z)

m(z)

2

dz

,

(33)

[M1] =

l1

0m1

2m(z) dz

l10

m12m(z)

m(z)dz

l10

m12m(z)

m(z)dz

l10

m1

m(z)

m(z)

2dz

,

(34)

[K2] =

l2

0EI2

a (x)

a(x)2

dx

l2

0EI2(

a (x))

2a(x) dx

l2

0EI2(

a (x))

2a(x) dx

l2

0EI2(

a (x))

2 dx ,(35)

[M2] =

m2l2 0

0

l20

m22a(x) dx

. (36)

Regarding the device as a series-connected massspring

damper system, the transformation of force and torque in this

system can be calculated using the equivalent circuit of the

impedance in figure5(b). The system works at the resonant

frequency and the mode is defined, thus, regardless of theangle,xcan be expressed byz. The system has one degree

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of freedom, but has a different displacement impedance in

different directions.

The proof mass, the main beam and the auxiliary beam

have the same generalized displacement at the joint point at

the free end of the auxiliary beam (point a in figure 5). As

such, the impedance of the proof mass and the impedance of

the main and auxiliary beams have the following condition:

[F]

2[Mp]=

[F][Z1][Z2]

[Z1]+[Z2]

, (37)

where [F

] is the force applied to the proof mass, 2[Mp]

is the impedance of the proof mass, [F]is the force applied

to the main beam and the auxiliary beam, and [Z1][Z2][Z1]+[Z2]

is the

impedance of the main beam and the auxiliary beam. Drawn

from figure 5(b), the {F} has infinite impedance; thus, the

force in the x-direction applied at the free end of the main

beam is:

Fx = (m2l2+ Mp)m1(l1)(2)

{F}

[Z0]

T

1

0

+

{F}

[Z0]

[Z1] [Z2]

[Z1] + [Z2]

T 10

. (38)

The torque Mthat is applied to the free end of the main

beam also excites the main beam; here,Mis:

M= Mpl2a1(l2)(2)

{F}

[Z0]

T01

+ {F}T

0

1

Mpl2.

(39)

The expressions for the force and torque show that they

result from two sources: the induced acceleration of the

motion of the proof mass and the inertial acceleration of the

base movement.

At low frequency, the impedance of the device is high;

thus, the induced acceleration of the motion of the device

is extremely low. In this case, the inertial force {F} applied

to the proof mass transfers the force ( {F}[Z0]

[Z1][Z2][Z1]+[Z2]

)T

10

,

as well as generates the torque {F}T

01

Mpl2 directly at

the main beam. However, at the initial resonant frequency,

the impedance of the device is extremely low, and {F}has a phase /2, which is different from the motion of

the device. Thus, the induced inertial force of the proof

mass generates the force (m2l2+Mp)m1(l1)(2)(

{F}[Z0]

)T

10

and torque Mpl2a1(l2)(

2)({F}[Z0]

)T

01

applied to the main

beam. Furthermore, the applied force and torque determine

the modes of the main beam. At the initial resonant frequency,

the force applied to the free end of the main beam Fxis

Fx = (m2l2+ Mp)m1(l1)(2)

{F}

[Z0]

T 10

. (40)

Here,Fx excites the main beam, and the torque Mapplied tothe free end of the main beam also plays the role of activating

the main beam and is given by

M= Mpl2a1(l2)(2)

{F}

[Z0]

T 01

. (41)

As the low rigidity auxiliary beam lowers the series

resonant frequency of the device and makes the main beam

work in a quasi-stationary state, the gradient of the strain

distribution of the main beam is mainly caused by the force

in the x-direction applied at the free end of the main beam.

On the other hand, the torque applied at the free end of the

main beam smooths out the strain distribution. Thus, the strain

distribution of the main beam is determined by the ratio of the

maximum torque generated by the force in thex-direction and

the torque directly applied at the free end of the main beam.

The ratio can be expressed as

Fxl1

M =

(m2l2+ Mp)m1(l1)(2)(

{F}[Z0]

)T

1

0

l1

Mpl2a1(l2)(2)({F}[Z0]

)T

0

1

m1(l1)l1

a1(l2)l2. (42)

Here,m1(l1)anda1(l2)can be simply treated as the relative

amplitude of the proof mass in the x-direction andz-direction.

As the rigidity of the auxiliary beam is reduced, the free end of

the auxiliary beam would inevitably have much larger motion

in the z-direction than in the x-direction. Hence, the induced

acceleration of the proof mass would be much larger in the

z-direction than in the x-direction. In addition, the torque

directly applied at the main beam would be much larger.Thus, the strain distribution of the main beam would also

be smoothed. The auxiliary beam with proof mass acts as an

absorber which transforms the x-direction movement into the

z-direction movement and provides a large torque at the free

end of the main beam. Thus, the strain distribution of main

beam is smoothed.

3.3. Voltage output

In this section, the voltage output across a resistance load is

studied. The constitutive equations of the strips are given by:

S1 = sE11T1+d31E3, (43)

D3 = d31T1+ T33E3, (44)

whereS1 andT1 are the strain and stress along the direction

of the piezoelectric strips, respectively; D3 and E3 are the

electrical displacement and the electric field, respectively; and

d31 and 33 are the compliance values at constant electric

field, that is, the piezoelectric strain constant and the dielectric

constant under constant stress, respectively.

Assuming the leakage resistance of the PZT is much

higher than the load resistance, which is usually insignificant

in an electrical circuit, the electrical part of the model issimplified as a resistive load R between two electrodes of the

7


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Table 1. Amplitude of the proof mass in the z-direction andx-direction. (Note: unit: mm.)

Device

Amplitude (excited byUz) Amplitude (excited byUx)

|w2(l2, t)| |w1(l1, t)| |w2(l2, t)| |w1(l1, t)|

Model I 10.21 0.54 0.91 0.05Model II 3.16 0.43 1.52 0.19Model III 1.9 0.57 0.91 0.27

Figure 6. Simplified circuit representation.

piezoelectric disc. The electric current output can be written

as follows:

d

dt

A

D n dA

=

v(t)

R, (45)

where

D is the vector of the electric displacement

components in the piezoelectric layer, and n is the unit

outward normal. As shown in figure 6, the relation betweenthe charge on the electrodes q and the voltage across the

piezoelectric discV(t)is obtained as follows:

1

Rx+

s33bl1

hp

dV

dt= e31b

hp

2 +a

l10

3w

z2t. (46)

Assuming thatV,D3and S1vary as ejt, then

V=e31b(

hp2 +a)

l10

3wz2t

1Rx

+jCpejt

= e31b(

hp2 +a)x(t)

l10 Sm(z) dz

1Rx

+jCpejt. (47)

This equation indicates that the piezoelectric elements

voltage output is proportional to the strain integral of the

piezoelectric element when the other parameters are kept

constant. Assuming that the strain distribution in the length

direction is equal to the strain distribution on the surface of

the beam, it can be concluded that the electrical energy stored

in the piezoelectric element is proportional to the square of

the strain integral. A smoothed-strain cantilever can produce

much more energy than an unsmoothed-strain cantilever undera strain limitation at the same resonant frequency.

4. Simulation and discussion

In this section, the strain distribution of three prototypes is

examined by simulation and experiment at the first resonant

frequencies of the devices, where they are most effective. The

three prototypes have auxiliary beams with different stiffness

levels. In this experiment, the length of the auxiliary beam is

half that of the main beam: thus, ki = EIaEIeq

is used to evaluate

the relative stiffness of the auxiliary beams in comparison with

the main beam.

Model I has ki = 0.02, which means that the stiffness ofthe auxiliary beam is much lower than that of the main beam.

Model II haski = 0.25.

Model III has ki = 3, which means that the stiffness of

the auxiliary beam is higher than that of the main beam.

The section5presents the parameter settings in detail.

As discussed in section 3.3, the ratio of the maximum

torque generated by the force in the x-direction and the

torque directly applied at the free end of the main beam is

mainly determined by the amplitude of the proof mass in the

x-direction and z-direction. The amplitude of the proof mass

in the x-direction andz-direction for the three devices, when

they are separately excited by thez-direction vibration and thex-direction vibration, are shown in table1.

Table 1 was obtained at damped free vibration at the

respective initial frequencies of each device using the FEM

method under a sinusoidal acceleration of 2.5 m s2. As the

rigidity of the auxiliary beam decreases, the amplitude of the

motion in the z-direction becomes much larger than that in

thex-direction. Thus, the torque applied at the free end of the

main beam is larger than the maximum torque generated by

the force in the x-direction, and the strain distribution of the

main beam is smoothed. For the low-stiffness Model I, the

stimulated resonance frequency is 22.1 Hz. The stimulated

resonance frequencies for Models II and III are 36 and

47.5 Hz, respectively.

As the three systems have different resonance frequen-

cies, the strain distribution at each resonance frequency

is studied by the FEM method; for better comparison,

normalized differences of strain on the surface of the PZT

element are shown with experimental data in section5.3.

5. Experiments and discussion

5.1. Experimental setup

In the experiment, right-angle cantilevers were fabricated

with the following dimensions: l1 = 30 mm, l2 = 15 mm,b = 20 mm, Mp = 53.4 g, hp = 0.4 mm, hb = 0.8 mm,

8


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Table 2. Vibration amplitude of the proof mass in the z-direction andx-direction. (Note: unit: mm.)

Device

Amplitude (excited byUz) Amplitude (excited byUx)

|w2(l2, t)| |w1(l1, t)| |w2(l2, t)| |w1(l1, t)|

Model I 11.5 0.62 0.95 0.05Model II 2.95 0.35 1.61 0.17Model III 1.6 0.52 1.01 0.31

Figure 7. Photograph of the experimental system.

p = 7.5 g c m3, b = 4.51 g cm

3, a = 7.8 g c m3,

Eb =109 GPa andEp =106 GPa. The experimental setup is

shown in figure7.First, the piezoelectric element was bonded

to the titanium (TA1) substructure layer. The ceramic material,

PZT-5, was purchased from Jiaye Company, China. It had a

charge coefficient of d31 = 171 pC N1. The brass proofmass, of size 20 mm15 mm20 mm, weighs 53.4 g. The

PZT and polished sheets were bonded using DP460 (epoxy

glue) and cured for 2 h at 70 C. This procedure was repeated

when the unimorph was bonded to the orthogonal steel lever.

The three auxiliary beams of different stiffness for

Models I, II and III were made of steel sheet. These auxiliary

beams were also of different thicknesses. The thickness

for Model I was 0.0003 m, with an estimated stiffness of

0.0049 N m2, less than that of the main beam, which was

0.243 N m2. The thickness for Model II was 0.0007 m, with

an estimated stiffness of 0.062 N m2, and that for Model III

was 0.0016 m, with an estimated stiffness of 0.74 N m2.

5.2. Vibration amplitude and strain distribution

In this experiment, vibration amplitudes of the proof mass in

the z-direction and x-direction were measured using a laser

sensor; strain distributions on the surfaces of the PZT sheets

during vibration were obtained by strain gages. The resistance

of the strain gages was 120 1 , with an active area

of 1.0 mm 1.0 mm. We used models with seven gages

along the mid-lines of the PZT element to measure the strain

distribution and models, with just one gage at the end of the

PZT element to measure the voltage output.

Table2 shows the measured vibration amplitudes of theproof mass in the z-direction and x-direction using a laser

Figure 8. Stimulated and measured strain distributions of the threemodels.

sensor under a sinusoidal acceleration of 2.5 m s2 at each

resonant frequency, which agree well with the stimulation

results.

Figure 8 shows the simulated and measured straindistributions on the surfaces of the PZT elements of three

models at each resonant frequency.

The simulated data are obtained for undamped free

vibration at the initial frequencies of each device, respectively.

It is apparent that a highly uniform strain distribution for the

low-stiffness model exists at its resonance frequency. A peak

strain exists at the clamped end of the main beam, due to

the boundary condition at the clamped end. As the stiffness

of the extended beam increases, the resonance frequency of

the model increases. The gradient of the strain distributions

of Models II and II show a increasing trend, compared with

the low-stiffness model. However, adding a dampener wouldslightly smoothen the strain distribution.

Models with seven gages along the mid-line of the PZT

element are used to measure the strain distribution at resonant

frequencies; the data are also shown in figure 8 and fit

the stimulated result well. The strain distribution is mainly

determined by the ratio of the maximum torque generated by

the force in the x-direction and the torque directly applied at

the free end of the main beam. When the auxiliary beam has

much lower stiffness, the auxiliary beam provides extremely

large torque, and the main beam has a highly uniform strain

distribution. As the stiffness of the extended beam increases,

the gradient of the strain distribution also increases. This isconfirmed by the strain distribution of Models II and III.

9


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Figure 9. RUE of the PZT element versus frequency.

5.3. RUE of the PZT element

Normally, a piezoelectric energy harvester (unimorph or

biomorph) works at its initial resonant frequency, and the

highest strain in the longitude direction occurs at the clamped

end. From the discussion in section3.3,the charges resulting

from mechanical strain promote the integration of the strain

distribution in longitude, when other parameters are kept

constant. The highest utilization efficiency of a PZT element

can be achieved only when the piezoelectric material has

a uniform strain distribution. In following discussion, a

standard RUE of a PZT element is set for evaluation; thestandard is based on the assumption that the longitudinal strain

distribution of a PZT element is absolutely uniform. Here,

the rectangular PZT element with constant width and height

is studied. In addition, the PZT element used in the current

study is covered with continuum electrodes without any

segmentation, and the influence of the electrode is ignored.

The RUE of the right-angle piezoelectric cantilevers can

be defined as follows:

=

l1

0 (zm) dzm

l10 |(zm)|maxdzm

. (48)

Based on FEM stimulation results, the frequency range is

much lower than the second resonant frequency, and the RUE

of a conventional rectangular piezoelectric cantilever with a

proof mass is 50%. Erturks study also supported the RUE of

a conventional cantilever [29].

Figure9shows the RUE at different frequencies for the

three models, excited by either the z-direction vibration or

the x-direction vibration, respectively. The simulations were

carried out in the open-circuit condition. Figure 9also shows

the value of the RUE of a conventional composite cantilever.

Model I. When the device is excited by the z-direction

vibration, the RUE of the PZT element approximates 97% in a

quasi-static activation. This is due to a torque directly appliedat the free end of the main beam, as discussed in section3.3.

The RUE of the PZT element decreases as the frequency

increases. When the device is excited by the x-direction

vibration, the RUE of the PZT element approximates that of a

rectangular conventional composite cantilever in a quasi-static

activation. This is because the auxiliary beam provides an

inertial force to the main beam, mainly because the device

works in a quasi-static state. The RUE increases as thefrequency increases. The two curves excited by the z-direction

vibration and x-direction vibration intersect at the initial

resonant frequency of the device, which is 22.1 Hz, where

the efficiency approximates 93%. The intersection of the two

curves agrees well with the EulerBernoulli beam theory.

Figure 9 also indicates that, when the device is excited by

the x-direction vibration, the frequency is higher than the

resonance frequency. The RUE of the PZT element has a

peak, due to the transition of shapes between different modes.

However, the device could not be used at these frequencies

because they are not the resonant frequencies, and the device

cannot be driven efficiently.

Model II. When the device is excited by the z-directionvibration, the RUE of the PZT element is equal to that of

Model I in a quasi-static state. The RUE of the PZT element

decreases as frequency increases at a faster rate. When the

device is excited by the x-direction vibration, the RUE of the

PZT element also equates to that of Model I in a quasi-static

condition. The RUE increases as the frequency increases. The

two curves excited by thez-direction vibration andx-direction

vibration intersect at the initial resonant frequency of the

device, at 36 Hz, where the efficiency approximates 87%. The

RUE of the PZT element also has a peak that is higher than the

initial resonance frequency when excited by the x-direction

vibration.Model III. The RUE of the PZT element approximates

the same, although a little lower than that of Model II when

the device is excited by the z-direction vibration. The RUE

of the PZT element shows a slower increase trend, compared

with that of Model II when the device is excited by the

x-direction vibration. The two curves of the device excited by

thez-direction vibration and x-direction vibration intersect at

the initial resonant frequency of the device, at 47.5 Hz, where

the efficiency approximates 80.5%.

Figure 9 indicates that the RUE of the PZT element

can be very high in a model that has a low stiffness of

the auxiliary beam. The increasing stiffness of the auxiliary

beam may be lower than the RUE at the resonant frequency

of the device. The experiments agree with the stimulation

well. Moreover, figure9 indicates the relative capacity of the

power output of the PZT element. The capacity of the power

output of a PZT element is proportional to the square of the

integration of the mechanical strain distribution in longitude at

a certain frequency. A high RUE of the PZT element indicates

a high capacity for power output at a certain frequency with a

determined strain limitation.

5.4. Voltage output

Given that a PZT plate with more uniform strain distributioncan undoubtedly have a greater potential voltage output,

10


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Figure 10. Strain limitation versus voltage output.

relationships between the output voltage and the maximumstrain of each system were tested and compared in this work.

The mode of the device at the initial resonant frequency and

the voltage output under the strain limit were also defined,

regardless of the directions of the vibration. Thus, the voltage

outputs under a strain limit were measured at each initial

resonance frequency, respectively. The devices were excited

by thez-direction vibration, and the strain gages were bonded

to the surfaces at the clamped ends of the main beams.

This comparison can sufficiently demonstrate the difference

between them.

Figure 10 shows the respective peak-to-peak voltages

generated by the PZT plates at the initial frequencies ofthe devices, with the maximum strain varying from 10 to

26 . Model I shows a maximum voltage output which

is 1.1 times more than that of Model II under a certain

strain limitation, and 1.2 times more than that of Model III.

Notably, the power output of the devices is much larger than

that of a conventional device with the same strain limitation.

In an earlier work, Baker described the advantages of a

strain-smoothed cantilever[16].

Furthermore, the performances of the devices in two

special cases were tested; they were excited by the x-direction

vibration andz-direction vibration.

The peak-to-peak voltage outputs of the three models

shown in figure 11 were obtained under a sinusoidal

acceleration of 2.5 m s2 exerted on the base. Model I reached

its open-circuit peak-to-peak voltage output of 602.7 V at

22.4 Hz, which was excited by the z-direction vibration. When

the cantilever was excited by the x-direction vibration, the

open-circuit voltage output reached its peak of 49.7 V at

22.6 Hz. The voltage output under the z-direction activation

was 12 times that under the x-direction activation. This was

due to the forcedisplacement impedance of the device in

the z-direction, which was lower than that in the x-direction.

The device exercised a larger distance in the direction of the

base movement as well as absorbed and transformed more

energy. The results agree well with the analysis presented insection3.3.

Figure 11. Open-circle voltage versus vibration frequency.

Additionally, Model II reached its open-circuit peak-to-

peak voltage output of 468.6 V at 36.5 Hz, when excited

by the z-direction vibration; when excited by the x-direction

vibration, the voltage reached a peak of 203.2 V at 36.7 Hz.

The device generated 2.3 times more voltage output when

excited by the z-direction vibration than when excited by the

x-direction vibration.

Model III was excited by the z-direction vibration. The

open-circuit peak-to-peak voltage output reached its peak of

326.4 V at 48 Hz. When excited by the x-direction vibration,

the voltage reached a peak of 272.4 V at 48.3 Hz.

Moreover, the frequencies at the peak of the voltage

output were slightly different when the device was excited

by the x-direction vibration and z-direction vibration. The

frequency at the peak of the voltage output was slightly lower

when the device was excited byz-direction vibration than that

by thex-direction vibration. This is due to the effect of gravity

and rotation of the base.

5.5. Power output

This section studies the power output of the three models. To

investigate the power output of the right-angle cantilever, aresistive load is added to the output of the piezoelectric plate.

Only the frequency response around the first natural frequency

was studied. The generator was excited with a sinusoidal

vibration of 2.5 m s2.

Figure12shows the output voltage versus load resistance

and output power versus load resistance, in which the best

match load is a function of frequency. Hence, although the

piezoelectric elements of the three devices have the same size,

their best match loads are different. Best fit resistance loads

can be chosen from figure 12for the devices, namely, 670,

340 and 290 k for Models I, II and III, respectively. The

resulting peak-to-peak voltage Upp across the load resistancewas measured by an oscilloscope. The peak energy was

11


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Figure 12. (a) Output voltage versus load resistance, (b) output power versus load resistance.

Figure 13. Power output versus frequency.

calculated using the following equation:

Woutput =V2pp

8Rload.

Figure13shows the power outputs of three models with

the same vibration excitation of a sinusoidal acceleration of

2.5 m s2. Model I reached its maximum power output at

22.3 Hz, which was excited by the z-direction vibration. The

output power was 20.12 mW. When excited by the x-direction

vibration, the power reached a peak of 22.3 Hz, with a

power output of 0.17 mW. Model II reached its maximum

power output of 9.3 mW at 36.3 Hz, which was excited bythe z-direction vibration. When excited by the x-direction

vibration, the maximum peak-to-peak output power reached

2.39 mW at 36.4 Hz. When Model III was excited by the

z-direction vibration, the power output reached a peak of

5.7 mW at 47.9 Hz. Finally, when excited by the x-direction

vibration, the power reached a peak of 4.82 mW at 48 Hz.

6. Conclusion

Piezoelectric elements can be attached to mechanical

structures and used to transform vibration energy to

electrical energy. The present paper proposes and analyzes

a right-angle cantilever that can dramatically smoothen thestrain distribution of the piezoelectric plate. The right-angle

cantilever has an auxiliary beam that can provide a large

torque to the main beam; the strain distribution on the

main beam is highly uniform. Hence, the RUE of the PZT

element is considerably heightened and the performance of

the cantilever is greatly improved.The RUE of a PZT element in a conventional rectangular

piezoelectric cantilever with a proof mass is 50%. However,

the right-angle structure can dramatically improve it at the

initial resonant frequencies of the devices, and the RUE

increases as the stiffness of the auxiliary beam decreases. The

RUE of a PZT element is 93% when the auxiliary beam has

a stiffness which is 0.02 times that of the main beam; it is

87% when the auxiliary beam has a stiffness which is 0.25

times that of the main beam; and 80.5% when the auxiliary

beam has a stiffness which is three times that of the main

beam. The voltage outputs and power outputs of three models

under two directions of activation were also studied. Undera strain limitation, the device with higher RUE can generate

higher voltage. Furthermore, the device with higher RUE can

also generate more voltage and power under certain base

movements in thez-direction.

In summary, the right-angle structure can greatly

smoothen the strain distribution of the main beam and

considerably enhance the RUE of the PZT plate, thus

dramatically improving the voltage and power outputs. The

analysis and stimulations are consistent with the experiments.

Acknowledgment

This work was supported by the Fundamental Research Funds

for the Central Universities.

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Smart Materials and Structures Volume 21 Issue 6 2012 [Doi 10.1088_0964-1726_21!6!065017] Xu, Jia...

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