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Composite Structures

journal homepage: www.elsevier.com/locate/compstruct

A novel composite multi-layer piezoelectric energy harvester

Qingqing Lua,c, Liwu Liub, Fabrizio Scarpac,d,⁎, Jinsong Lenga, Yanju Liub,⁎

a Centre for Composite Materials, Science Park of Harbin Institute of Technology (HIT), P.O. Box 3011, No. 2 YiKuang Street, Harbin 150080, People’s Republic of ChinabDepartment of Astronautical Science and Mechanics, Harbin Institute of Technology (HIT), P.O. Box 301, No. 92 West Dazhi Street, Harbin 150001, People’s Republic ofChinac Bristol Composites Institute (ACCIS), University of Bristol, Bristol BS8 1TR, UKd Dynamics and Control Research Group (DCRG), CAME, University of Bristol, BS8 1TR, UK

A R T I C L E I N F O

Keywords:Piezoelectric energy harvestingComposites beamVibration energy harvester

A B S T R A C T

A typical linear piezoelectric energy harvester (PEH) is represented by a unimorph or bimorph cantilever beam.To improve the efficiency of linear PEHs, classical strategies involve the increase of the beam length, tapering oradding additional cantilever beams to the free end. In this work we discuss the design of novel type of compositelinear multi-layer piezoelectric energy harvester (MPEH). MPEHs here consist of carbon fibre laminates used asconducting layers, and glass fibre laminas as insulating components. We develop first a electromechanical modelof the MPEH with parallel connection of PZT layers based on Euler-Bernoulli beam theory. The voltage and beammotion equations are obtained for harmonic excitations at arbitrary frequencies, and the coupling effect can beobtained from the response of the system. A direct comparison between MPEH and PEH configurations is per-formed both from the simulation (analytical and numerical) and experimental point of views. The experimentsagree well with the model developed, and show that a MPEH configuration with the same flexural stiffness of aPEH can generate up to 1.98–2.5 times higher voltage output than a typical piezoelectric energy harvester withthe same load resistance.

1. Introduction

The development of wireless sensor networks and the reduction oftheir power requirements has significantly driven research activitiesrelated to vibration piezoelectric energy harvesting technologies, whichpossess high energy conversion efficiencies from mechanical vibrationsto electrical power, all within the use of simple designs [1]. In 1996,Williams and Yates first introduced the concept of vibration energyharvesting [2]. Wang et al. (1999) then presented the constitutiveequations of symmetrical triple layer piezoelectric benders, and in-troduced different types of piezoelectric benders, like unimorphs andbimorphs with series or parallel connection [3]. Erturk and Inman haveestablished the fundamental electromechanical model of a piezoelectriccantilever beam, with their 2008 papers related to a single-degree-of-freedom (SDOF) harvester beam with one PZT layer [4,5]. The sameAuthors then investigated the analytical model of bimorph cantileverconfigurations with series and parallel connections of PZT layers [6].Micro-fibre composites (MFC) can also be used in vibration energyharvesters. Song et al. developed the theoretical model for energyharvesting devices with two types of MFC materials, and the influenceof the beam thickness, natural frequency and electrical resistance were

also experimental investigated [7]. Nonlinear PEHs have been ad-ditionally designed and developed in recent years to expand the op-erational frequency bandwidth. Erturk et al. [8,9] have presented anonlinear broadband piezoelectric power generator with two magnetsnear the free end of the cantilever beam and with two piezoelectriclayers on the surface. The piezo-magnetoelastic configuration proposedcould generate a power one order of magnitude larger than the oneprovided by a linear PEH at several frequencies. The performance of anonlinear magnetopiezoelastic energy harvester driven by random ex-citations was described by Litak et al. [10] and Ali et al. [11]. Theirwork shows that it is possible to optimally design the system by usinganalytical techniques, such that the mean harvested power is max-imized for a given strength of the input broadband random ambientexcitation. Ferrari et al. utilized two layers of PZT films to fabricate apiezoelectric bimorph, and the resulting nonlinear converter proposedimplements nonlinearity and bistability by using a single externalmagnet [12]. Friswell et al. have also designed a new type of highlynonlinear piezoelectric cantilever beam, exhibiting two potential wellswith large tip masses, when the beam is buckled [13]. Other nonlinearPEHs with magnets can be found in Refs. [14–16], and in the reviewarticle [17]. Betts et al. presents an arrangement of bistable composite

https://doi.org/10.1016/j.compstruct.2018.06.024Received 2 March 2018; Accepted 6 June 2018

⁎ Corresponding authors at: Bristol Composites Institute (ACCIS), University of Bristol, Bristol BS8 1TR, UK (F. Scarpa).E-mail addresses: f.scarpa@bristol.ac.uk (F. Scarpa), yj_liu@hit.edu.cn (Y. Liu).

Composite Structures 201 (2018) 121–130

0263-8223/ © 2018 Published by Elsevier Ltd.

T

plates with bonded piezoelectric patches to perform broadband vibra-tion-based energy harvesting from ambient mechanical vibrations [18].Arrieta et al. have designed a cantilevered piezoelectric bi-stablecomposite concept for broadband energy harvesting with two piezo-electric layers attached on the surface of bistable composites [19]. Panet al. used bistable hybrid symmetric laminates to make a broadbandpiezoelectric energy harvester, and have discussed the influence of thelay-up design on the performance of the bistable PEH [20,21]. Qi et al.have designed a multi-resonant beam with piezoelectric fibre compo-sites (PFC) to produce a broadband PEH [22]. A double cantilever en-ergy harvester was evaluated via a distributed parameter modelling andexperimental tests by Rafique et al. [23]. Adhikari et al. have used astack configuration piezoelectric energy harvester, and generatedpower from broadband vibration [24]. Paknejad et al. have presented adistributed parameter electroelastic model for various multilayercomposite beams and discussed the influence of the composites layups[25]. Akbar et al. have studied the dynamic response of piezoelectricenergy harvesters embedded in a wingbox structure [26]. Some newpiezoelectric energy harvester configurations have also been presentedto improve the efficiency during vibration, such as the L-shape [27], theV-shape [28] and the compressive-mode energy PEHs [29].

From the above literature review it is possible to observe that bothlinear and nonlinear PEHs are in general designed using a symmetricaltriple layer structure (i.e., one mid-structure with two piezo-layers onthe surfaces). This baseline configuration has been modified by usingdifferent beam/plate lengths or shapes (tapering), however the funda-mental layout remains the same. In this study we propose a PEHstructure with a more complex through-the-thickness topology. Theelectromechanical model of the composite multi-layer piezoelectricenergy harvester (MPEH) is here established based on the use of theEuler-Bernoulli beam theory. The fundamental natural frequency andthe power output of the MPEH beam are then extracted, and the per-formance of the MPEH is discussed based on simulations and experi-mental data. A parametric analysis of the voltage density and resonantfrequencies is also performed versus different stacking sequences of theMPEHs and PEHs leading to different specific flexural stiffness (i.e.,normalized by the mass of the beams). From the simulations, experi-mental data and the parametric analysis we will demonstrate that theMPEHs can generate significantly more power than the PEHs config-urations, in particular for small specific flexural stiffness configurations.The variations of the natural frequencies in MPEHs and PEHs with si-milar specific flexural stiffness are less pronounced, with the PEHshaving larger fundamental natural frequencies compared to the MPEHscases.

2. Analytical electro-mechanical model

Fig. 1 presents the design of the MPEH, which consists of multiplePZT layers and composites laminates. The MPEH is a cantilever beamwith carbon fibres, glass fibres laminates and PZT layers. The length ofcantilever beam is L, and b is the width. Fig. 1(b) and (c) represent thecross section of the MPEH, h h,ci cj are the thickness of the carbon fibrelaminates, hgi is the thickness of the glass fibre laminate. The thicknessof the PZT layer is hp, and hs is the total thickness of the beam except forthe external layers PZT. In the design, the polarization direction for allthe PZT layers is the same, which means that each pair of PZT layers areconnected in parallel. The carbon fibre laminates are here used asconducting layers, and the glass fibre laminates constitute the in-sulating layers.

2.1. Basic equations and fundamental natural frequency

The thickness is relatively small than the length of the compositebeam, so the general governing equation of motion of an Euler-Bernoulli beam with embedded piezoelectric layers can be written as[6],

∂∂

+ ∂∂ ∂

+ ∂∂

+ ∂∂

= − + ∂∂

M x tx

c I w x tx t

c w x tt

m w x tt

m M w x tt

( , ) ( , ) ( , ) ( , )

( ) ( , )

s a

tb

2

2

5

4

2

2

2

2 (1)

In Eq. (1), M x t( , ) is the bending moment of the beam; w x t( , ) is thebeam transverse deflection relative to its base, and w x t( , )b is the baseexcitation motion on the beam along the z-direction. The tip mass atx= L is Mt. The bending moment can be related to the internal stressesof the layers in the following manner:

∑ ∑ ∑

∫ ∫ ∫

∫ ∫ ∫

= − − −

= − − −= = =

M x t b σ zdz b σ zdz b σ zdz

b σ zdz b σ zdz b σ zdz

( , ) t c t g t p

k

N

t ck

N

t gk

N

t p1 1 1

c g p

c

ck

g

gk

p

pk (2)

In Eq. (2), σc, σg, σp correspond to the normal stress of carbon andglass fibre laminates, and of the piezoelectric layers along the x-axis,respectively. For the kth layer of the elastic composite laminate, thebending stress is defined:

= − ∂∂

σ σ z Q wx

( , ) ( )c g k k112

2 (3)

The kth layer stiffness Q11 can be defined by using the traditionalclassical laminate theory [30]:

= + + +=

=

=

=

−

−

−

Q Q θ Q Q θ θ Q θQ

Q

Q

Q G

cos 2( 2 )sin cos sinEν ν

ν Eν νEν ν

11 114

12 662 2

224

11 1

12 1

22 1

66 12

112 21

12 212 21

212 21

(4)

where E1 is the Young’s modulus along the fibre direction, E2 is theYoung’s modulus along the transverse direction, and G12 is the shearmodulus. The Poisson’s ratios are ν12 and ν21, and θ is the fibre or-ientation.

The stress in the PZT layer can be expressed by the constitutiveequations for a piezoelectric slab:

= −σ E ε x t e E t( , ) ( )p pp

1 31 3 (5)

In Eq. (5) Ep is the elastic modulus of piezoelectric layer, ε p1 is the

axial strain components, e31 is the piezoelectric constant and E3 is theelectric field across the thickness of the beam.

From the Euler-Bernoulli beam theory, the internal bending mo-ment can be obtained by substituting Eqs. (3), (5) into Eq. (2):

= − ∂∂

+ − −M x t YI w x tx

H x H x L( , ) ( , ) ϑ [ ( ) ( )]p2

2 (6)

where YI is the flexural rigidity of the piezoelectric elastic composites.The Heaviside function H(x) limits the location of the piezoelectriclayer along the x-direction on the host structure. The flexural rigiditycan be described by:

∫ ∫

∫

∑ ∑

∑

= +

+

= =

=

− −

−

YI b Q z dz b Q z dz

E b z dz

( ) ( )k

N

kc

z

z

k

N

kg

z

z

pi

N

z

z

111

2

111

2

1

2

c

k

kg

k

k

p

k

k

1 1

1 (7)

For a parallel connection the polarization direction is the same,therefore the e31 coefficient has the same sign for the top and bottomlayers. The directions of e31 and E t( )3 are expressed in Fig. 2, with blackand redarrows separately.

The piezoelectric coupling term ϑ can be derived from Eq. (2),which is a function of time only. The internal moment in PZT layer is:

Q. Lu et al. Composite Structures 201 (2018) 121–130

122

∫ ∫ ∫

∫

∫ ∫

∫

∫

⎜

⎟

= − ⎛⎝

+ +⋯+

+ ⎞⎠

= − ⎡⎣⎢

− +

− +⋯+ −

+ − ⎤⎦⎥

= − ⎡⎣

+ −

+⋯+ + − ⎤⎦

= + + +⋯+

+

− −

− +

− −

−

+

− −

−

+

− −

−

∗

∗ + +

( )( )

( ) ( )( ) ( )

M x t b σ zdz σ zdz σ zdz

σ zdz

b E ε x t e E t zdz E ε x t

e E t zdz E ε x t e E t zdz

E ε x t e E t zdz

M x t v t h

v t h

M x t v t v t v t

( , )

( ( , ) ( )) ( ( , )

( )) ( ( , ) ( ))

( ( , ) ( ))

( , ) ( )

( )

( , ) ϑ ( ) ϑ ( ) ϑ ( )

p hcih

ph

p hsh

p

h p

hcih

pp

hp

p

hsh

pp

h pp

pbeh p

i hp

h

pj h

ph

p pi

pi

pi

pi

pn

pn

2 2

21 31 3 1

31 32

1 31 3

1 31 3

22

2

2

2

2

2

2

2

1 1

hcip

hcp

hci hs p

hs p

hs

hcip

hcip

hci

hs p

hs p

hs

pci ci

s s

21

2

2 2

2

2

2

2

2

2

2

2

31

(8)

Substituting Eq. (6) into (1), the governing equation of the systemcan be obtained as:

∂∂

⎡⎣⎢

∂∂

⎤⎦⎥

+ ∂∂

⎡⎣⎢

∂∂ ∂

⎤⎦⎥

+ ∂∂

+ ∂∂

+ − − + ⋯+ − −

= − + ∂∂

xYI w x t

x xc I w x t

x tc w x t

tm w x t

t

v t H x H x L v t H x H x L

m M w x tt

( , ) ( , ) ( , ) ( , )

ϑ ( )[ ( ) ( )] ϑ ( )[ ( ) ( )]

( ) ( , )

s a

pi

pi

pn

pn

tb

2

2

2

2

2

2

3

2

2

2

2

2 (9)

According to the linear piezoelectric materials constitutive equa-tions, another essential governing equation is necessary to determinethe unknown variables of w x t( , ) and vp as:

= +D e ε x t ε E( , )p S3 31 1 33 3 (10)

In Eq. (10), D3 is the electrical displacement and ε S33 is the permit-

tivity component at constant strain with the plane-stress assumption( = −ε ε d ES T

p33 33 312 ).

The electric current can be obtained by applying Guess’ law [31].Since the resistance across the inner and outer PZT layers is R, the totalcurrent can be expressed as:

∫+ ⋯+ =⎯⇀⎯( )v

RvR

ddt

D n dA·pi

pn

A 3 (11)

where⎯⇀⎯n is the unit outward normal, and the integration is performed

over the electrode area.Since = − ∂

∂ε x t z( , )p

pw x t

x1( , )22 , by combining Eqs. (10) and (11) one

obtains:

∫

∫

⎧

⎨

⎪

⎩⎪

+ = −

⋮

+ = −

∂∂ ∂

∂∂ ∂

e bz dx

e bz dx

vR

bLεh

dv tdt p

io

L w x tx t

vR

bLεh

dv tdt p

no

L w x tx t

( )31

( , )

( )31

( , )

pi p

p

i

pn p

p

n

33 32

33 32 (12)

where zp is the distance between the centre of the PZT layers and theneutral axis. We can then solve Eqs. (9) and (12) by assuming the beamdeflection with the modal expansion:

Fig. 1. Design of multi-layer piezoelectric energy harvester (MPEH). (a) Vibration cantilever beam with multi-layer composites laminates and PZT layers; (b) Theinternal design of the MPEH; (c) Cross section of a MPEH.

Fig. 2. Polarization direction and electrical field of parallel connection.

Q. Lu et al. Composite Structures 201 (2018) 121–130

123

∑==

∞

w x t ϕ x η t( , ) ( ) ( )r

r r1 (13)

where ϕ x( )r is the mass normalized eigenfunction of the rth vibrationmode and η t( )r is the modal participation factor. The mode ϕ x( )r can beobtained by solving by undamped free vibration problem with clamped-free boundary conditions:

= + + +ϕ x C λ x C λ x C λ x C λ x( ) cosh sinh cos sinr r r r r1 2 3 4 (14)

where Ci is an amplitude constant and λr is the eigenvalue, which canbe obtained from solving the characteristic equation with the BCs:

= ′ =

′ = ′ = −′ ′′

ϕ ϕ

YIϕ L YIϕ L ω M ϕ L

(0) 0, (0) 0

( ) 0, ( ) ( )r r

r r r t r2

(15)

The eigenvalues λr can be calculated from the classical transcen-dental equation:

= +−

ω MYIλ

λL λLλL λL λL λL1 cosh cos

cosh sin sinh cosr t2

3 (16)

The rth natural frequency can be therefore expressed as follows:

= = …ω λ YIm

r 1, 2, 3r r2

(17)

Eq. (16) can then be rewritten as follows:

− − + =M λm

λL λL λL λL λL λL(cosh sin sinh cos ) (1 cosh cos ) 0t(18)

The main vibration mode function can be therefore expressed as= − − −ϕ x C λx λx A λx λx( ) (cosh cos (sinh sin ))1 .

According to Ref. [6], = ⎛

⎝⎜

⎞

⎠⎟

− + −

+ − −A

λL λL λ λL λL

λL λL λ λL λL

sin sinh (cos cosh )

cos cosh (sin sinh )

MtmMtm

.

The constant C1 can be evaluated by applying the orthogonalityconditions:

∫

∫

∫∫

⎡⎣⎢

⎤⎦⎥

= −

+ ⎡⎣⎢

⎤⎦⎥

−

=

ϕ x ddx

YId ϕ x

dxdx ω M ϕ L ϕ L

d ϕ xdx

YId ϕ x

dxdx

YId ϕ x

dxd ϕ x

dxdx ω M ϕ L ϕ L

ω mϕ x ϕ x dx

( )( )

( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )

Ls

rr t r s

L s r

L r sr t r s

rL

r s

0

2

2

2

22

0

2

2

2

2

0

2

2

2

22

20 (19)

when ≠r s:

∫ ∫= + =M ϕ L ϕ L mϕ x ϕ x dx YId ϕ x

dxd ϕ x

dxdx0 ( ) ( ) ( ) ( ) ,

( ) ( )0.t r s

Lr s

L r s0 0

2

2

2

2

In the case =r s: ∫+ =M ϕ L m ϕ x dx[ ( )] [ ( )] 1t rL

r2

02 ,

∫ ⎡⎣

⎤⎦

=YI dx ωL d ϕ xdx r0

( ) 22r

2

2 .

2.2. Output voltage and relative displacement

The equation of motion of the multi-layer PEH in modal coordinatescan be obtained by substituting Eq. (13) into Eq. (9):

+ + + + ⋯+ =d η t

dtζ ω

dη tdt

ω η t χ v t χ v t f t( )

2( )

( ) ( ) ( ) ( )rr r

rr r r

i irn n

r

2

22

(20)

In Eq. (20), the model electromechanical coupling term is:

= ==

χdϕ x

dxi nϑ

( ), 1:r

ipi r

x L (21)

The model damping ratio is ζr . The model force can be expressed as:

∫= −⎡⎣

⎤⎦

−f t ϕ x mdxd g t

dtM ϕ L

d g tdt

( ) ( )( )

( )( )

rL

r t r0

2

2

2

2 (22)

According to Kirchhoff laws, the relationship between the voltageand the current can be obtained as:

⎧

⎨⎪

⎩⎪

+ =⋮

+ =

C i t

C i t

( )

( )

pdv t

dtv t

Ri

pdv t

dtv t

Rn

( ) ( )2

( ) ( )2

i i

n n

(23)

If the PZT layers have all the same resistance, then Eq. (23) can beexpressed as

+ + + = +C d v t v tdt

v t v tR

i t i t[ ( ) ( )] ( ) ( )2

( ) ( )pi n i n

i n(24)

where:

∑ ∑= = ==

∞

=

∞

Cε bL

hi t κ

dη tdt

i t κdη t

dt, ( )

( ), ( )

( )p

S

p

i

rri r n

rrn r33

1 1

The model coupling term is then:

= −

⋮

= −

+

=

+

=

κ

κ

ri e h h b dϕ x

dx x L

rn e h h b dϕ x

dx x L

( )2

( )

( )2

( )

c p r

s p r

31 1

31

(25)

The base excitation is assumed as a harmonic vibration around thez-direction:

==

g t Y ef t F e

( )( )

jωt

r rjwt

0

(26)

The model force Fr can be given by Eqs. (26) and (22):

∫= − −F mω Y ϕ x dx M ϕ L Y ω( ) ( )rL

r t r2

0 0 02

(27)

For the harmonic vibration case, the voltage and the model me-chanical response are assumed to be harmonic at the same frequency:

==

η t H ev t Ve

( )( )r r

jωt

jωt (28)

where the temporal term amplitude Hr and V are calculated by sub-stituting Eqs. (27) and (28) into Eqs. (20) and (24).

∑

∑

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

+ − =

⋮

+ − =

− + + + +⋯+ =

=

∞

=

∞

( )

( )

jωC V jω κ H

jωC V jω κ H

ω j ζ ω ω ω H χ V χ V F

0

0

( 2 )

R pi

rri

r

R pn

rrn

r

r r r r ri i

rn n

r

12

1

12

12 2

(29)

From Eq. (29), one can obtain Hr and V i, and then substitute in Eq.(28) to express the voltage output of the system:

∑

∑=

+ +

=

∞

− +

=

∞+ ⋯ +

− +

v tjωC

e( )i r

jωκ Fj ζ ω ω ω ω

R pr

jω κ χ κ χj ζ ω ω ω ω

jωt12

12

1

( )2

ri r

r r r

ri i r

n n

r r r

2 2

2 2(30)

An important point to be drawn from observing Eq. (30) is that theeffect of the embedded additional PZT layers does not consist only inadd another PEH with a parallel connection; the voltage output of the

Q. Lu et al. Composite Structures 201 (2018) 121–130

124

inner and outer part of the beam has also a coupling influence.The steady state mechanical response of the multi-layer PEH can be

given as follows:

=

− + +∑ + ⋯ +

+=

∞η t F

j ζ ω ω ω ω

e( )

2

rr

r r r

jω κ χ κ χ

jωC

jωt

2 2( )

rri i r

n n

R p

11

2 (31)

The displacement at point x is therefore:

∑=

− + +∑=

∞

+ ⋯ +

+=

∞w x tF ϕ x

j ζ ω ω ω ω

e( , )( )

2r

r r

r r r

jω κ χ κ χ

jωC

jωt

12 2

( )r

ri i r

n n

R p

11

2 (32)

Furthermore, the total power output provided by the multi-layerPEH is:

=

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

+⋯

+⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

∑

∑

∑

∑

+ +

+ +

=

∞

− +

=

∞+ ⋯ +

− +

=

∞

− +

=

∞+ ⋯ +

− +

P t

e

e

R( )jωC

jωt

jωC

jωt

2

2

r

jωκri Frj ζr ωr ω ω ωr

R pr

jω κri χi κrnχn

j ζr ωr ω ω ωr

r

jωκrnFrj ζr ωr ω ω ωr

R pr

jω κri χi κrnχn

j ζr ωr ω ω ωr

1 2 2 2

12

1

( )2 2 2

1 2 2 2

12

1

( )2 2 2

(33)

2.3. MPEH with 4 PZT layers

From the model presented in Sections 2.1 and 2.2, one can calculatethe fundamental natural frequency and the voltage output of MPEH. Ifthe MPEH has four PZT layers (Fig. 3), then the model equations can befurther simplified.

In that case, the flexural rigidity of the piezoelectric elastic com-posites of Eq. (7) can be expressed as:

∑ ∑ ∑

∑ ∑

∫ ∫ ∫

∫ ∫

= + +

= ⎡

⎣⎢ +

+ + − + + − ⎤

⎦⎥

= = =

= =

− − −

− −

( )( ) ( ) ( ) ( )

YI b Q z dz b Q z dz E b z dz

b Q z dz Q z dz

h h

( ) ( )

( ) ( )

k

N

kc

zz

k

N

kg

zz

pi

N

zz

k

N

kc

zz

k

N

kg

zz

E hp

h hp

h

111

2

111

2

1

2

111

2

111

2

23 2

3

2

3

2

3

2

3

c

kk

g

kk

p

kk

c

kk

g

kk

p c c s s

1 1 1

1 1

1 1

(34)

The internal bending moment exist in the PZT layer can be obtainedfrom Eq. (8):

∫ ∫ ∫

∫

∫ ∫

∫ ∫

∫ ∫

∫ ∫

⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎜ ⎟

⎜ ⎟

= − ⎛

⎝⎜ + +

+ ⎞

⎠⎟

= − ⎡

⎣⎢ +

+ +

+ +

+ − + − ⎤

⎦⎥ =

−⎡

⎣

⎢⎢⎢

+ −

−⎤

⎦

⎥⎥⎥

=

− ⎡

⎣⎢

⎛⎝

⎛⎝

+ ⎞⎠

−⎛⎝

⎞⎠

⎞⎠

− ⎛⎝

⎛⎝

⎞⎠

−⎛⎝

+ ⎞⎠

⎞⎠

+ ⎛⎝

⎛⎝

+ ⎞⎠

−⎛⎝

⎞⎠

⎞⎠

− ⎛⎝

⎛⎝

⎞⎠

− + ⎞⎠

⎤

⎦⎥

= − ⎡

⎣⎢

⎛⎝

⎛⎝

+ ⎞⎠

−⎛⎝

⎞⎠

⎞⎠

+ ⎛⎝

⎛⎝

+ ⎞⎠

− ⎞⎠

⎤

⎦⎥ = + +

+ +

− −

−

− −

−

+ +

− −

−

− −

−

+ +

− −

−

− −

− ∗

+ +

− −

−

− − −

∗

∗

∗

M x t b σ zdz σ zdz σ zdz

σ zdz

b E ε x t zdz E ε x t zdz

E ε x t zdz E ε x t zdz

ev t

hzdz e

v th

zdz

ev t

hzdz e

v th

zdz M x t

b ev t

hz e

v th

z ev t

hz

ev t

hz M x t

bev t

hh h h v t

hh h h

v th

h h h v th

h h h

M x t beh

v t h h h

v t h h h M x t v t v t

( , )

( , ) ( , )

( , ) ( , )

( ) ( )

( ) ( )( , )

( ) ( ) ( )

( )( , )

( )2 2

( )2 2

( )2 2

( )2

(2

)

( , ) 2 ( )2 2

( )2

(2

) ( , ) ϑ ( ) ϑ ( )

p hc

hc hpp hs

hs hpp hc hp

hc

p

hs hp

hs

p

hc

hc hpp

phs

hs hpp

p

hc hp

hc

pp

hs hp

hs

pp

hc

hc hp pin

phs

hs hp pout

p

hc hp

hcpin

phs hp

hspout

pp

pin

p hc

hc hppout

p hs

hs hppin

p hc hp

hc

pout

p hs hphs p

pin

p

cp

c pin

p

c cp

pout

p

sp

s pout

p

s sp

pp

pin c

pc

pout s

ps

p pin

pin

pout

pout

12

12

2

21

2

12

2

2

12

12

12

21

12

12

12

21

12

12

312

231

12

12

312

231

312

12

12

312

2

2

312

12

12

312

2 2

311

21

21

21

2

2 2 22

31 12

12

22

(35)

In Eq. (35), ∗M x t( , )p is the internal moment induced by the elasticpart of the PZT layer, ϑp

in is the piezoelectric coupling term of two innerPZTs and ϑp

out is the piezoelectric coupling term of two outer PZT layers.The output voltage of the inner and outer PZT layers are vp

in and vpout .

3. Experimental setup

The hybrid MPEH fabricated consists of carbon fibre laminates(SE70 CFRP, Gurit Ltd), glass fibre composites (SE70 E-Glass, Gurit,Ltd) and PZT (Sinoceramics Inc., Shanghai, China) layers. The lowcuring temperature of the prepreg (70 °C) is significantly lower than theCurie one of the PZT. The properties of prepregs are listed in Table 1.The plies are sized by using a cutting machine (Genesis 2100, BLACK &WHITE Ltd), then laid up with a PZT PZT[ /0 /90 /0 /0 /90 / /90 ]c c g c c c Sstacking sequence. Here the subscript c indicates the CFRP, and g standsfor the E-glass layer. The PZT layers are placed inside the hybridcomposites plies before curing. The resin inside the prepregs also acts asadhesive for the PZT layers. The curing progress is performed in anoven at °70 C, under a pressure of 1 bar for 16 h. Here, the pressure iscontrolled by a vacuum pump which connected with the vacuum-bagged specimen. After curing the plate was cut into a 10mm widthand 95mm length. The dimensions of the PZT layers connected inparallel are × ×10 mm 80 mm 0.2 mm. Copper foils are placed on thefree surfaces of the PZTs, and connected with copper wires as elec-trodes. The tip mass fixed at the free end is a steel rectangular block of

× ×10 mm 10 mm 15 mm (11.7 g).To validate the electro-mechanical model, the fundamental natural

frequency and the power generation ability of the MPEH are firstlytested. The experimental setup is shown in Fig. 4. The MPEH sample is

Fig. 3. Cross-section of MPEH with 4 PZT layers.

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held by two thick acrylic plates and mounted on the top of a shaker(LDS shaker system), which is controlled by a NI USB-6221 multi-functional I/O board (National Instrument) and power amplifier (LDSPA25E Shaker Power Amplifier). The sensors are an accelerometer (PCB333M07) and single point laser vibrometer (Ometron VPI 8330). Thesignal of the time domain velocity has been then FFT-transformed, witha sampling frequency of 2048 Hz and sampling time of 3 s. The signalhas been subjected to a Hanning window to reduce the spectral leakage.The accelerometer is fixed to the clamped side of the beam to monitorthe vibration of shaker, and the laser vibrometer measures the velocityof the tip mass.

The natural frequency is firstly measured by using an impacthummer (force transducer PCB 086D05) while the MPEH is fixed at theclamp. The output of the force transducer and the laser vibrometer isgenerated through the NI USB-6221 from a signal coupler (Kistler5134).

To measure the power generation performance, the MPEH is fixed tothe shaker as a clamped-free beam. The multifunction I/O device gen-erates a sinusoidal waveform to the power amplifier. The amplitude ofthe vibration acceleration is regulated through the amplifier ratio, and

the power generation is measured at different acceleration levels. Theoutput voltage is measured by an oscilloscope (PicoScope 2204A) withtwo channels. Two channels measure the voltage of two inner PZTlayers and the outer two PZT layers, respectively.

The theoretical electromechanical model presented above has beencoded in a Matlab platform. The frequencies used in both the experi-mental and numerical model range between 50 Hz and 115 Hz duringthe output voltage test at 10 kΩ resistance. Several different excitationaccelerations (0.5 g, 1 g, 2 g, 3 g) are chosen to discuss the performanceof the MPEH. The output power is calculated based on the experimentaldata under different resistance loads as =P V R/2 .

For further benchmarking, a finite element (FE) model is also de-veloped using the commercial code ABAQUS. The composite laminatesare represented by solid laminate elements (C3D8R), while the PZTlayers are simulated by using piezoelectric element C3D8E (8-nodelinear piezoelectric brick). The FEA mesh discretization is representedby squares with 2.5 mm side for the composites beam, and bricks with2.5 mm side for the PZT transducers. The surfaces of the PZTs arebonded to the surfaces of the composites beam by ‘tie’ constraint duringthe assembly. The FEA eigensolver is a Lanczos one.

4. Results and discussions

4.1. Fundamental frequency of the MPEH

The results of the natural frequency associated to the first mode areshown in Table 2. The experimental result is 91.9 Hz, with an averagemodal damping ratio calculated from three FRFs as =ζ 0.018. Thetheoretical model provides a 92.6 Hz natural frequency. The finiteelement analysis result (91.07 Hz) is also close to the experimental testdata.

Table 1Parameters of materials.

Properties SE70 CFRP SE70 E-Glass

PZT-5H

Density [kg/m3] ρ 1502 1936 7500Thickness [mm] t 0.2 0.2 0.2Moduli [GPa] E1 130.33 40.74 61

E2 7.22 10.1 61E3 – – 48G12 4.23 3.82 23G13 – – 23G23 – – 23.5

Poisson’s ratios v12 0.337 0.255 0.289v13 – – 0.512v23 – – 0.408

Piezoelectric strain constants[×10−12 m/V]

d d,31 32 – – -270d33 – – 550

Dielectric property [×10−8 F/m] D11 – – 1.505D22 – – 1.505D33 – – 1.301

Fig. 4. Experimental setup for the vibration test of the MPEH system. (1) Computer for data logging and control; (2) NI USB-6211 multifunction I/O device; (3)Kistler 5134 power supply/coupler; (4) Ling Dynamic Systems LDS PA25E Shaker Power Amplifier; (5) Electrodynamic shaker; (6) MPEH with tip mass; (7)Acceleration transducer; (8) Ometron scanning laser Doppler vibrometer 8330; (9) Vibrometer controller; (10) PicoScope 2004A oscilloscope.

Table 2Natural frequency.

Natural Frequency Error

Theoretical model 92.6 Hz 0.76%FEA 91.07 Hz 0.9%Experimental test 91.9 Hz –

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4.2. Simulation results of voltage response

The simulation results of MPEH with different resistance loads areshown in Fig. 5. The maximum voltage FRF value is close near thenatural frequency, with the voltage getting larger with the increase ofresistance R. In this case the load resistances are increased from 470Ω to

K20 Ω, and the peak values of the first mode are listed in Table 3. Thevoltage FRF amplitude at 470 Ω is 1.003 V/g, while at 20 kΩ the voltageFRF increases to 7.7 V/g. The corresponding frequency of the peakvalue shifts with different load resistances. For small Rs (470 Ω), theresonance frequency is 89.9 Hz, which increases slowly to 90.9 Hz at

=R 5 kΩ. After that point the resonance frequency visibly increases to93.3 Hz at =R 10 kΩ, then the increase is less significant (94.9 Hz at20 kΩ). The same type of behavior is also observed by Rafique et al. inthe case of bimorph cantilever energy harvesters [23].

We have made a comparison between the proposed MPEH designand a traditional PEH with only two PZT layers on the external surfaces.

Firstly, both the MPEH and the PEH have the same length, width,thickness, tip mass and materials. The layup of PEH is the following:PZT[ /0 /90 /0 /0 /90 /0 /90 ]c c g c c c c S. The stacking sequences difference be-tween the MPEH and PEH only exists in the second ply, the inner PZTlayers of the MPEH are replaced by 0 degree CFRP plies. The flexuralstiffness of composites beam can be calculated by the follow equation:

∑ ⎜ ⎟= ⎛⎝

+ ⎞⎠

= +=

−D Q t zt

z z z( )12

, 12

( )k

N

K k kk

k k k111

112

3

1(36)

Here Q11 is defined in Eq. (4). The flexural stiffness of the MPEH andPEH with 0.2mm thickness PZT can be calculated: =D 223 N/mMPEH11 ,

=D 225 N/mPEH11 . We can find that the flexural stiffness of the MPEHand the PEH are almost the same, the error is only 0.9%.

Fig. 6 shows the comparison of the voltage FRFs between the MPEHand the PEH with two different load resistances. One can observe thatthe resonant frequencies of the PEH are higher than the ones of theMPEH. The MPEH amplitude of the voltage response is however sig-nificantly improved compared to the PEH case (Table 3). For a re-sistance of 5 kΩ, the voltage FRF of the MPEH reaches 3.52 V/g at90.9 Hz, while the PEH can only generate 1.4 V/g at 99.7 Hz. For eachload resistances used, the maximum voltage generated by the MPEH onaverage is about 2.2 times higher than the PEH.

We also consider the case in which the thickness of the PZT layersused in the PEH is increased to 0.4mm, which means that the totalvolume of the PZTs for the PEH and the MPEH is the same. In that case,if all the PZT layers are located on the external surfaces of the com-posites beam, the flexural stiffness D11 term will become larger than theone of the MPEH configuration. The resonance frequency of the PEHwould be significantly higher than the MPEH one, and the powergenerated by the PEH would be smaller than the MPEH case (Fig. 7).The MPEH and the PEH with 0.4 mm thickness PZTs are calculated

Fig. 5. Voltage FRF theoretical simulation for the MPEH.

Table 3Comparisons between maximum voltage FRFs of the MPEH and the PEH.

Resistance MPEH PEH

Maximumvoltage FRF(V/g)

Correspondingfrequency (Hz)

Maximumvoltage FRF(V/g)

Correspondingfrequency (Hz)

470Ω 1.003 89.97 0.37 98.091 kΩ 1.91 90.13 0.65 98.092 kΩ 2.05 90.13 0.91 98.215 kΩ 3.52 90.92 1.40 99.6810 kΩ 4.17 93.31 1.92 100.515 kΩ 5.58 94.11 2.66 101.320 kΩ 6.50 94.9 3.38 102.9

Fig. 6. Comparisons of the voltage FRF between the MPEH and the PEH.

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without tip mass.A more comprehensive comparison between the MPEH and PEH

configurations (all based on using 0.2mm thickness PZTs) is by con-sidering their resonant frequencies and voltage density FRFs versus thespecific flexural stiffness (D m/11 , where m is the mass of the compositebeams). The stacking sequences of the beams are changed to obtaindifferent flexural stiffness D11. The volumes of the beams for all thestacking sequences are kept the same ( × ×8 1 0.32 cm3), as well as themodal damping ratio, for simplicity. As shown in Fig. 8, the peak

voltage density FRF decreases with the increase of the specific flexuralstiffness, but the values provided by the MPEHs are always larger thanthe ones provided by the PEHs. This is particularly valid for low specificflexural stiffness (D m/11 less than −e15 N/m/kg3 ). These results are quiteimportant, because they show that by using MPEHs with comparablemore compliant stacking sequences, the output voltage density is sig-nificantly enhanced compared a PEH configuration. The difference interms of resonant frequencies between MPEHs and PEHs is however lessmarked, and the two configurations show a very similar sensitivitytrend versus the specific flexural stiffness.

4.3. Experimental validation

Fig. 9 shows the experimental results related to the multi-layerpiezoelectric energy harvester with a 10 kΩ load resistance under fourdifferent base accelerations. The maximum RMS voltage occurs at92 Hz. The analogous peak in the theoretical model is 93.3 Hz. For anexcitation of 0.5 g the RMS voltage of the MPEH is 2.24 V; with theincrease of the excitation level, the voltage generated by the MPEHreaches 3.79 V at 1 g, 7.61 V at 2 g and 12.07 V at 3 g. The averagemaximum experimental voltage FRF is 4.02 V/g, which is close to thetheoretical result of the MPEH with the same load resistance (4.17 V/g).The average error between the experimental data and the simulationresults at 10 kΩ is 3.57%.

The sensitivity of the MPEH performance versus different load re-sistance values is shown in Fig. 10. The voltage FRF at 470 Ω is 0.71 V/gand occurs at 91 Hz. The analogous theoretical result is 1.003 V/g at89.97 Hz. The frequency slowly increases to 91.3 Hz at 5 kΩ. When the

Fig. 7. Voltage FRF comparisons between the MPEH and the PEH (0.4 mm thickness PZT).

Fig. 8. Peak voltages density FRFs and resonant frequencies versus the specificflexural stiffness D m/11 for the MPEHs and the PEHs.

Fig. 9. Experimental RMS voltages with 10 kΩ load resistance of the MPEH.

Fig. 10. Experimental voltage responses of MPEH at seven difference loads.

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load resistance is 20 kΩ the resonance frequency moves to 92.5 Hz. Inthe range between 470 Ω and 20 kΩ, the experimental results show ageneral good agreement with the ones provided by the model devel-oped in this work. The output voltage of the MPEH becomes larger withthe increasing load resistance, and the resonance frequency of themaximum voltage FRF also shift towards higher values.

Fig. 11 features a comparison between experimental and simulatedresonance frequencies and voltage FRFs related to MPEHs and PEHs. Inthis case, the resonance frequency corresponds to the frequency atwhich the magnitude of the FRF is the largest. The PEH considered inthis part is fabricated with the same materials used for the MPEH, andthe stacking sequence is PZT[ /0 /90 /0 /0 /90 /0 /90 ]c c g c c c c S. Both the simu-lations and the experimental data show that the resonance frequenciesfor the two types of energy harvester increase with the load resistances.The maximum percentage error between experimental and theoreticalresonance frequencies is 2.52%. For a given resistance, the MPEH cangenerate more voltage than the PEH configuration. At 5 kΩ the MPEHcan produce a 2.5 times higher voltage FRF peak than the PEH, and at20 kΩ resistance the MPEH can generate again about 1.98 times morevoltage than the PEH system. Fig. 11 also shows that the theoreticalmodel can reproduce the response of the MPEH system in a very ade-quate fashion.

The output power FRFs of the MPEH is shown in Fig. 12. Themaximum power for a given load resistance occurs at the same re-sonance frequencies illustrated in Fig. 10. With lower resistances (lessthan 10 kΩ), the maximum amplitude has peak at 1 kΩ (2.66mW/g),after which the power decreases with the increase of the resistance(1.62 mW/g at 10 kΩ). For increasing loads (15 kΩ or 20 kΩ), the powerresponse shows a gradual increase to 2.18mW/g.

5. Conclusions

In this paper, a new design of piezoelectric energy harvesting systemwith multiple composite laminates and PZT layers is presented. Theelectromechanical model of the MPEH is developed based on Euler-Bernoulli beam theory. The structure of the MPEH consists of carbonfibre and glass fibre laminates, and PZT layers. The elastic properties ofthe beams are defined by using classical laminate theory, and the effectgiven by the parallel connection of PZT layers is also taken into ac-count. The model can reproduce the fundamental frequencies of theMPEH system and the voltage response with different load resistances.The results from the simulations show that the MPEH voltage FRF ismaximized when the excitation frequency is close to the one associatedto the first mode of the MPEH. The resonance frequency of the max-imum voltage output changes with the load resistances, which indicatesthat an increase of the load resistances makes the resonance frequencylarger. Both the experimental and the simulation results give evidencethat the voltage FRFs of the MPEH increases with larger load re-sistances, and the modal provides a very good comparison with the testdata. The comparison between the MPEH and a classical PEH config-uration shows that the MPEH can provide an output voltage between1.98 and 2.5 times higher of the PEH system with the same load re-sistance. It is therefore apparent that multi-layer piezoelectric energyharvesting designs can generate more power compared to traditionalones. MPEHs can therefore be considered for further concepts, like themodification of MEMS PEH, or nonlinear vibration applications cur-rently covered by PEH designs.

Acknowledgements

QL acknowledges the support of the National Natural ScienceFoundation of China (Grant Nos.: 11632005, 11672086, 11421091)and China Scholarship Council. FS is grateful to the logistic supportfrom the MSCA ITN VIPER program regarding the use of the softwareand the access to the Material Laboratory of the Faculty of Engineering.QL and FS are also grateful for the access to BLADE (Bristol Laboratoryof Dynamics Engineering) of the University of Bristol for the dynamicstests.

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