199
© Institution o Engineers Australia, 2008
* Reviewed paper srcinally presented at the FirstWorld Congress on Engineering Asset Management(1
st
WCEAM), Gold Coast, Australia, 2006.† Corresponding author Dr Ian Howard can becontacted at I.Howard@curtin.edu.au.
Computational FEA model of a coupled piezoelectricsensor and plate structure for energy harvesting
*
MF Lumentut, KK Teh and I Howard
†
Department o Mechanical Engineering, Curtin University o Technology
SUMMARY:
This paper presents a mathematical model o a piezoplate energyharvestingscheme. An analytical method is used to generate a fnite element model o the coupled piezoelectricsensor element using LoveKirchho’s plate theory. Constitutive equations or a single layer plate element are ormulated. The polarisation o the piezoelectric sensor bounded on the upper plate structure is due to ambient vibration exerted on the structure. Forced vibration o the smartstructure will create strain energy within the crystalline structure o the piezoelectric material.The resulting electric feld generated by the sensor element was modelled using a linear thicknessinterpolation unction and the meshed plate elements were modelled using ournode rectangularelements with three degrees o reedom or each node. The structural eigenmodes and dynamicresponse o the coupled piezoplate system were solved by using modal analysis and Newmarkintegration methods respectively. The analysis is demonstrated with both dynamic displacementand electric voltage responses to an applied step orce. Further modelling o the smart structure isaimed at maximising the power generation capability.
F orce, NKE Kinetic Energy, JPE Potential Energy, JPEE electrical energy, JK stiness matrix, N/mM consistentmass matrix, kg
Greek Letters
strain vectors
σ
stress vectors, N/m
2
dielectric matrix at constant strain, F/m
poison ratio
shape unction
z
( )
electric potential, Volts
nodal vectors, m
dierential operator o shape unction
p
density o plate, kg/m
3
s
density o sensor, kg/m
3
total o energy, Nm
NOMENCLATURE
u displacement o plate element inx direction, mv displacement o plate elementin y direction, mw displacement o plate element inz direction, mC modulus o elastic, N/m
2
D electric displacement vectors, C/m
2
d piezoelectric constant, m/Ve piezoelectric stress coefcient, C/m
2
E electric feld, V/mD
m
stiness coefcient, N/m
2
t thickness, ml length, mw width, m
Australian Journal o Mechanical Engineering, Vol 5 No 2
200“Computational FEA model o a coupled piezoelectric sensor and plate …” –
Lumentut, Teh & Howard Australian Journal o Mechanical Engineering Vol 5 No 2
Ω work done o the system, J
Subscripts
x relative to xaxisy relative to xaxisz relative to zaxisn reer to joints o plate element
θθ
reer to plate
Φθ
reer to sensor–plate
θΦ
reer to platesensor
Φ
reer to sensor
θ
reer to plate
Superscripts
s reer to sensor elementp reer to plate elementps reer to platesensor elementT matrix transpose
1 INTRODUCTION
The investigation o piezoelectric elements as smartstructures has many wideranging applications inthe engineering industry. The ability o piezoelectricmaterials to transorm mechanical strain, induced by ambient vibration, to electrical potential is wellknown, as evidenced in the direct mode by using thematerial as a sensor. On the other hand, piezoelectricmaterial which has the ability to transorm electricalfeld to mechanical strain is also wellknown, as theconverse mode o using the material as an actuator.Numerical analysis o these smart structures usinga three dimensional fnite element model was frstpresented by Allike and Hughes in 1970 (Allik,1970). However, a three dimensional fnite elementanalysis o the integrated piezo and plate structurewas challenging and generally demanded largecomputational requirements. It was realised thatthe piezoelectric patches were too thin and twodimensional modeling o these patches was sufcient.Other numerical models using laminated rectangularplates with piezoelectric material bonded on thesurace have been previously developed (Moita et al,2004; Tzou & Tzeng, 1990; Fernandes & Pouget, 2003),with other researchers using multilayer compositeplates (Saravanos et al, 1996; Kogl & Bucalem, 2005).Several publications, (Sekouri et al, 2004; Liu et al,2002; Taciroglu et al, 2004) have reported the use o numerical models to simulate the dynamic responseo the coupled elastic and electric feld o the smartstructures.The previous scenarios, mostly, discussed static anddynamic systems using the piezoelectric sensor andactuator bounded on the structures. Moreover, recentdevelopments o piezoelectric technology involve theusage o sensors patched on the structure subject toambient mechanical vibration. The induced strainenergy can be converted to useul electrical energycapable o being stored on electrical devices such as batteries or capacitors. Such a technique is generallyreerred to as the energy harvesting technique. Thedevelopment o energy harvesting techniques orpowering smart structures and embedded sensorshas received increased attention over the past decade(Sodano et al, 2004). Analytical and experimentalanalyses have been used to investigate powerharvesting rom PZT elements to power electricdevices or or recharging o batteries. EmbeddedPZT materials in a vibrating machine environmentcan be used as the required power source providedthe vibration source does not stop. A more useuldevelopment would be or the PZT element tostore its energy into a rechargeable battery or lateruse. Further analytical and experimental studieswith optimising power ow or adaptive energyharvesting and selpower harvesting have beendeveloped
(Ottman et al, 2002; James et al, 2004;Roundy et al, 2003; GlynneJones, 2004).The main objective o this paper is to presenta mathematical model o an energy harvestingtechnique with bonded piezosensors on a plateelement. Mechanical strain energy inducespolarisation in the piezosensor thus creating anelectric feld. The mathematical model introducesa ournode nonconormed quadrilateral elementwith a total o twelve nodal degrees o reedom,ormulated using LoveKirchho’s plate theory.The paper presents results rom a twosided fxedfxed plate using numerical algorithms solved usingMATLAB, based on the suggested ormulation.
2 CONSTITUTIVE EQUATIONOF PLATE ELEMENT
Classical plate theory can be used to derive theequations o motion or plates by assuming thatthe shear deormation eect is negligible. Thedisplacement vectors
u
,
v
, and
w
o an arbitrary pointin the deormed element can be written in terms o the midsurace o the plate, as described in fgure1a and fgure 1b and equations 1, 2 and 3.
u
=
z
w
x
,(1)
v
=
z
w
y
,(2)
w
=
w
(
x
,
y
)
(3)
201“Computational FEA model o a coupled piezoelectric sensor and plate …” –
Lumentut, Teh & Howard Australian Journal o Mechanical Engineering Vol 5 No 2
xw
∂∂
xw z u
∂∂−=
Figure 1a:
The deormations o plate elementwith respect.
yw
∂∂
yw z v
∂∂=

Figure 1b:
The deormation o plate elementwith to x axis respect to y axis.The corresponding straindisplacement vectorrelationship in the x and y directions, can beormulated as,
{ }
=
x
y
xy
=
z
2
w
x
2
2
w
y
2
2
2
w
x
y
.(4)A state o plane stress is assumed and the correspondingstressstrain relationship is,
{ }
=
C
1
v
2
1
v
0
v
1 00 01
v
2
x
y
xy
,(5)or
{ }
=
D
m
{ }
.(6)
3 CONSTITUTIVE EQUATIONOF PIEZOSENSOR
The piezoelectricsensor has a direct mode electricallyon its material due to the exerted mechanical strainon the structure. The matrix equation relating themechanical and electrical quantities can be writtenas (Naillon et al, 1983),
D
{ }
=
e
{ }
+
E
{ }
, (7)where
D
{ }
,
e
=
d
D
m
,
{ }
,
,
d
,
D
m
and
E
{ }
are the electric displacement vector, piezoelectricstress coeicient, strain vector, dielectric matrixat constant strain, piezoelectric constant, stinesscoefcient and electric feld vector respectively. Inthis case, the converse mode gained by the actuatoris neglected.Discritised electric feld {E} induced by ambientvibration generates polarisation in the piezosensormaterial, in the z direction along the sensor platethickness. The subsequent electrical potential isassumed linear and is ormulated as,
z
( )
{ }
=
z
( )
s
z
( )
s
{ }
,(8)where
z
( )
s
=
z
t p
2
t s
is the shape unction over theinterval t
p
/2 < z < t
p
/2 + t
s
.The electrical feld is a unction o the electricalpotential with negative gradient operator,
E
{ }
=
z
( )
{ }
=
z
( )
s
z
( )
s
{ }
,(9)where
is a gradient operator, frst derivative o theshape unction with respect to thickness direction,giving,
z
( )
s
=
0 0
d
z
( )
s
dz
T
=
0 01
t
s
T
.(10)
4 FINITE ELEMENT FORMULATION4.1 Discretised element plate matrix
As mentioned previously, the classical plate theorycan be urther dealt with by establishing a 12termpolynomial unction to model the nonconormingtransverse displacement,
w
(x,y)
.
The three degrees o reedom at each node
w
n(x,y)
,
θ
xn
and
θ
yn
can then beormulated as shown in fgure 2.The nodal degrees o reedom or the ournodedrectangular element, can be expressed in the vectororm, {
n x
,
y
( )
p
}
{
i x
,
y
( )
p
,
j x
,
y
( )
p
,
k x
,
y
( )
p
,
l x
,
y
( )
p
} and
{
w
n,
θ
xn,
θ
yn
}. All variables o displacement obtainedwill be substituted into a polynomial unction wherethe results will give the consistent displacementunction as,
202“Computational FEA model o a coupled piezoelectric sensor and plate …” –
Lumentut, Teh & Howard Australian Journal o Mechanical Engineering Vol 5 No 2
w
x
,
y
( )
=
n
(
x
,
y
)
p
n x
,
y
( )
p
{ }
,(11)where
n x
,
y
( )
p
is the shape unction o thedisplacement feld or the nonconorming plateelement with reerence to transversal displacementunction,
n x
,
y
( )
p
=
n
x
,
y
( )
p
1
,
n
{
i,j,k,l
} .(12)
(xy)
w
Zx
tsd
PlatePiezosensor
dx,ady, b
t
2
p
p
t
y jik
θ
x
θ
y
Figure 2:
Geometry element o the coupledpiezosensor and plate element.The straindisplacement relationship rom equation(4) can be expanded by orming the second orderpartial dierential equation o shape unction,equation (12), with respect to x and y axes o theelement plate in each joint as equation (13).Equation (13) can be simply rewritten as ollows,
¿¾½¯®»»»»»»»»»»»»¼º««««««««««««¬ª
∂∂
»¼º«¬ª
Φ∂∂∂
»¼º«¬ª
Φ∂∂∂
»¼º«¬ª
Φ∂∂∂
»¼º«¬ª
Φ∂∂
»¼º«¬ª
Φ∂∂
»¼º«¬ª
Φ∂∂
»¼º«¬ª
Φ∂∂
»¼º«¬ª
Φ∂∂
»¼º«¬ª
Φ∂∂
»¼º«¬ª
Φ∂∂
»¼º«¬ª
Φ∂∂
»¼º«¬ª
Φ∂−=
¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© §
p y xn y x p y xl y x p y xk y x p y x j y x p y xi y p y xl y p y xk y p y x j y p y xi x p y xl x p y xk x p y x j x p y xi z
,
į
,22,22,22,222,22,22,22,22,22,22,22,2
ε
.
(13)
=
– z
n x
,
y
( )
p
n x
,
y
( )
p
{ }
,
n
{
i,j,k,l
}.(14)
4.2 Governing differential dynamic equations
By invoking Hamilton’s principle or a conservativedynamic system and applying Lagrange’ssimplifcation, the general dynamic equation can beexpressed as,
d dt
u
.
u
F
=
1
s
= 0, (15)where,
= KE – PE + PEE,
.
n x
,
y
( )
p
n x
,
y
( )
p
{ }
,
z
( )
s
( )
; F
η
{{Fs},{Fc},{
q
}}The kinetic energy term is given by,
KE
=
12
p
w
.
x
,
y
( )
T
w
.
x
,
y
( )
d
(
vol
)
+
12
s
w
.
T
w
.
x
,
y
( )
d
(
vol
)
(16)and the potential energy can be written,
PE
=
12
{ }
T
{ }
d(vol)
. (17)The electrical energy term can be ormed as,
PEE
=
12
E
{ }
T
D
{ }
d
(
vol
)
. (18)Work done on the system is due to the surace orce,concentrated orce and electrical charge density andcan be written as,
=
w
x
,
y
( )
{ }
T
Fs
{ }
d area
( )
+
w
x
,
y
( )
{ }
T
Fc
{ }
w
x
,
y
( )
{ }
T
q
{ }
d area
( )
.(19)The solution procedure involves substitutingequations (6), (7), (9) and (14) into related expressionso equations (16) to (19) and dierentiating withrespect to all displacement vector parameters o the coupled sensor and plate structure. As given inequation (14), the resulting nonhomogenous matrixdierential equation or the smart structure can beormed as,
M
psn x y pzs
ÈÎ
»½
ÈÎÍÍÍ
»½¼¼¼
ÏÌÔÓÔ¸
( )( )
00 0
d d
..,..
¿¿
Ô
À
Ô+ÈÎ
»½
ÈÎ
»½
ÈÎ
»½
ÈÎ
»½
ÈÎÍÍÍÍ
»½
K K K K
T
ff fq qf qq
¼¼¼¼¼
ÏÌÔÓÔ¸
¿
Ô
À
Ô=ÏÌÔÓÔ¸
¿
Ô
À
Ô
( )( )
d d
f q
n x y pzs
F F
,
(20)where:
M
ps
=
i
n p x
,
y
( )
T
n p x
,
y
( )
d
(
vol
i
)
i
=
1
,(21)
203“Computational FEA model o a coupled piezoelectric sensor and plate …” –
Lumentut, Teh & Howard Australian Journal o Mechanical Engineering Vol 5 No 2
K
=
z
2
n p x
,
y
( )
T
D
i
n p x
,
y
( )
d
(
vol
i
)
i
=
1
,(22)
K
=
K
T
=
z
2
n p x
,
y
( )
T
e
T
z
( )
s
d
(
vol
s
)
s
=
1
,
(23)
K
=
z
( )
s
T
s
z
( )
s
d
(
vol
s
)
s
=
1
,(24)
F
{ }
=
n p x
,
y
( )
T
Fs
i
{ }
d
(
area
i
)
+
n p x
,
y
( )
T
Fc
i
{ }
i
=
1
,
(25)
F
{ }
=
z
( )
s
T
q
i
{ }
d
(
area
i
)
i
=
1
(26)To obtain the eigenmodes, dynamic displacementand electric voltage, the matrix dierential equation(20) can be solved using Guyan Reduction toseparate the degrees o reedom o the variabledisplacements where the equations yielded reectmore appropriately the independent equations orelectric voltage and linear dynamic displacement.At this point, the Newmark
β
method was used tosolve or the dynamic response o the piezoplatevibration system.
4.3 Application of boundary condition
Equations (21) to (26) can be solved by using integralalgebra and by incorporating appropriate geometric boundary conditions. In view o the dierentgeometrical boundary conditions or the sensorsand the base plate, special attention must be takenwhen modiying equations (21) to (24) to reect theseconditions.
5 RESULT AND DISCUSSION5.1 Numerical example
A MATLAB program was written based on theproposed theory to predict eigenmodes, dynamicdisplacement, and electrical voltages under transientresponse conditions. The characteristics or the piezosensor using PZT PSI5A4E (Piezo Systems, INC) andthe plate are listed in table 1 as ollows.The example used a twosided fxedfxed rectangularplate with piezosensors bonded on the upper suraceas shown in fgure 3. The plate was subjected to anexternally applied dynamic step orce o 1 Newtonat node 2, at time t=0 sec.Figure 3 shows the mesh pattern or the piezoplatestructure, consisting o 16 elements (4x4) and 16elements (4x4) o the piezosensors. The Piezosensor was bonded over the upper plate surace.The numerical methods outlined above were usedto solve the resulting fnite element model usingthe Newmark
β
method with the application o theMATLAB program.
Figure 3:
Geometry o the coupled piezosensorand plate element.
Table 1:
Characteristic data.ItemPiezosensorPlateC (GPA)66 207
0.30.3
(kg/m
3
)78007870t (m)0.0002670.002d
31
(m/V)190e12–d
33
(m/V)390e12–
33
(F/m)1.602e8–l x w (m
2
)0.25 x 0.2 (element)1 x 0.8
5.2 Eigenmodes
Figures 4a to 4d show the eigenvectors correspondingto the frst our natural requencies o the compositeplatepiezo structure. The eigenvector correspondingto the natural requency o 11.22 Hz shown in fgure4a indicates clearly the frst transverse vibrationo a ixedixed plate. Figure 4b shows the irsttwisting vibration mode at 14.31 Hz. The next twoeigenvectors shown in fgures 4c and 4d indicatethe second bending and twisting modes at 27.20 Hzand 31.23 Hz respectively. To optimise the locationo the piezo patch on the plate, modal analysis can be used to understand the prediction o deected