Noori, A finite element model based on coupled refined high-order global-local theory for static analysis of electromechanical embedded shear-mode piezoelectric sandwich composite beams wi[r]
Trang 1Original Article
Numerical optimization of piezolaminated beams under static and
dynamic excitations
Rajan L Wankhadea,*, Kamal M Bajoriab
a Applied Mechanics Department, Govt College of Engineering Nagpur, Maharashtra, 441108, India
b Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai, 400076, India
a r t i c l e i n f o
Article history:
Received 16 June 2016
Received in revised form
23 February 2017
Accepted 20 March 2017
Available online 30 March 2017
Keywords:
Piezoelectric
Finite element method
Higher order shear deformation theory
Actuator and sensor
a b s t r a c t
Shape and vibration controls of smart structures in structural applications have gained much attraction due to their ability of actuation and sensing The response of structure to bending, vibration, and buckling can be controlled by the use of this ability of a piezoelectric material In the present work, the static and dynamic control of smart piezolaminated beams is presented The optimal locations of piezoelectric patches are found out and then a detailed analysis is performed using finite element modeling considering the higher order shear deformation theory In the first part, for an extension mode, the piezolaminated beam with stacking sequence PZT5/Al/PZT5 is considered The length of the beam is
100 mm, whereas the thickness of an aluminum core is 16 mm and that of the piezo layer is of 1 mm The PZT actuators are positioned with an identical poling direction along the thickness and are excited by a direct current voltage of 10 V For the shear mode, the stacking sequence Al/PZT5/Al is adopted The length of the beam is kept the same as the extension mechanism i.e 100 mm, whereas the thickness of the aluminum core is 8 mm and that of the piezo layer is of 2 mm The actuator is excited by a direct current voltage of 20 V In the second part, the control of the piezolaminated beam with an optimal location of the actuator is investigated under a dynamic excitation Electromechanical loading is considered in thefinite element formulation for the analysis purpose Results are provided for beams with different boundary conditions and loading for future references Both the extension and shear actuation mechanisms are employed for the piezolaminated beam These results may be used to identify the response of a beam under static and dynamic excitations From the present work, the optimal location of a piezoelectric patch can be easily identified for the corresponding boundary condition of the beam
© 2017 The Authors Publishing services by Elsevier B.V on behalf of Vietnam National University, Hanoi This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)
1 Introduction
Piezoelectric materials are often used in various structural
ap-plications, including structural health monitoring, precision
posi-tioning, aeronautical and mechanical structures Piezolaminated
composite structures play an important role in controlling the
response of structure to shape and vibration as these are relatively
lightweight, strong, more stiff, and capable of sensing and actuating
than that of regular composites Piezoelectric materials are able to
produce an electrical response when mechanically stressed which
is a direct effect, and inversely high precision stresses can be
obtained with the application of an electric field Hence this property of sensing and actuation is used for the effective active shape, vibration, and buckling control To achieve the significant response of the structure in shape and vibration control, a piezo-electric material with different modes can be considered Intelli-gent (smart) structures and systems have become an emerging research area that is multi-disciplinary in nature, requiring tech-nical expertise from mechatech-nical engineering, structural engineer-ing, electrical engineering, applied mechanics, engineering mathematics, material science, computer science, biological sci-ence, etc The technology of smart structures is quite likely to contribute significant advancements in the design of high-performance structures, adaptive structures, high-precision sys-tems and micro/nano-mechanical syssys-tems This emerging area has been rapidly gaining momentum in the last few decades Also it can
be accepted that to some extent only initial, but highly feasible
* Corresponding author.
E-mail address: rajanw04@gmail.com (R.L Wankhade).
Peer review under responsibility of Vietnam National University, Hanoi.
Contents lists available atScienceDirect Journal of Science: Advanced Materials and Devices
j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / j s a m d
http://dx.doi.org/10.1016/j.jsamd.2017.03.002
2468-2179/© 2017 The Authors Publishing services by Elsevier B.V on behalf of Vietnam National University, Hanoi This is an open access article under the CC BY license
Journal of Science: Advanced Materials and Devices 2 (2017) 255e262
Trang 2studies of the concepts of piezolaminated structures have been
conducted[1] A mathematical model based on a layerwise theory
is also developed for laminated composite beams with embedded
piezoelectric actuators The one-dimensional beam formulation
also accounts for lateral strains for which thefinite element model
has been developed[2] Application of the piezoelectric material to
beams with considering Timoshenko vs EulereBernoulli theory
was performed to study the distributed control[3] The interaction
between active and passive vibration control characteristics was
investigated numerically using thefinite element approach These
numerically obtained characteristics were further verified
experi-mentally for carbon/epoxy laminated composite beams with a
collocated piezoceramic sensor and an actuator It is observed that
when the gain in velocity feedback control is small, the active
control follows the trend of the passive control, but provides
additional effects due to the active control Also for a large feedback
gain, the active control is dominant over the passive control[5]
A model simulating the effects of the control potential on the
static configuration of a piezo-elastic structure has also been
developed It is centered on the divergence free electric
displace-ment field [6] The active vibration control under damping was
investigated for the EulereBernoulli model with transverse
vibra-tions of a cantilever beam It was extended to include viscous and
KelvineVoigt (strain rate) damping Displacement and velocity
feedback controls provided by full length or patch piezoelectric
actuators and sensors bonded to the top and the bottom of the
beam can be used to control the response of the beam[7] A method
forfinding distributed actuators or dense actuator networks such
that a desired displacementfield is tracked was investigated
Dy-namic deflection tracking of beams by a distributed actuation and
the use of dense networks of actuator patches with a section-wise
constant intensity to approximate the effectiveness of the
distrib-uted actuators were also studied[8] The under critical frequency
behaviors related to deformed beams, showing the transition from
softening to hardening effects, were studied for various levels of
active voltage, static response and imperfection amplitudes of
simply-supported sandwich piezoelectric beams[9] Free vibration
and stability analysis of piezolaminated plates using the finite
element method employing the higher order shear deformation
theory were also performed Control of structure with its stability
was examined for simply supported smart piezolaminated
com-posite plates[10] Piezolaminated plates with cross ply and angle
ply stacking sequences with both symmetric and anti-symmetric
lay ups were studied[11,12] Theories for the accurate simulation
of the shear-mode behavior of thin or thick piezoelectric sandwich
composite beams have been developed considering both the
elec-trically induced strain component and the transverseflexibility of
structure [13] Vibrations of a cantilever piezolaminated beam
greatly influenced the extension and shear actuation mechanisms
of piezo actuators Also the proper placement of a piezo actuator
changes the response of beams in vibration in extension mode as
compared to that of the shear mode actuation [14] Vibration of
FGM Piezoelectric Plate also can be controlled using the LQR
Ge-netic Search technique[15] Shape control and vibration analysis of
the piezolaminated plates subjected to electro-mechanical loading
have been performed to study the effect of actuator voltages[16]
Hence fabrication and characterization of PZT string based MEMS
devices are mostly useful for this purpose The micro-fabrication
and characterization of free-standing doubly clamped
piezoelec-tric beams based on Pd/FeNi/Pd/PZT/LSMO/STO/Si heterostructures
were conducted, in which the displacements in static and dynamic
modes were investigated for string based MEMS devices [17]
In this work, thefinite element modeling for the extension and
shear modes of actuation for piezolaminated beams are considered
for shape control of the piezolaminated beams Further optimal
location of the piezoelectric actuator is considered for shape con-trol Different boundary conditions are considered in the piezola-minated beam analysis The optimal location of the piezoelectric actuator is obtained when stuying the response
2 Equilibrium equations Equilibrium equations are obtained using the virtual displace-ment principle Equilibrium between the internal and external forces has to be satisfied IfJrepresents the vector of the sum of the internal and external forces, then
where {R} represents the external forces due to imposed load and {P} is a vector of the internal resisting forces The equilibrium state
is achieved when {J}¼ 0 Further, {J} can be written as
fJg ¼ fRg þ1
2 Z
V
fεgTfsgdV 1
2 Z
V a
EpaTfDagdV
1 2 Z
V s
EpsTfDsgdV þ
Z
V
n
εNoT
fs0gdV (2)
where V, Vsand Vaare the area of the entire structure, the sensor layer, and the actuator layer, respectively Considering the work done by external forces due to the applied surface traction and applied electric charge on an actuator, the equation for the external work done can be written as
fRg ¼ Z
A
fugTfsðx; yÞgdA þ
Z
A
f0aqeðx; yÞdA (3) Thus the internal potential energy can be written as
P¼1 2 Z
V
deT½BT½C½B
de
dV1 2 Z
v a
deT½BT½eT½Ba
fe dV
1 2 Z
V s
deT½BT½eT½Bs
fes dV
1 2 Z
V a
feT½BaT½e½B
de
dV1 2 Z
V s
fesT½e½b
de dV
1 2 Z
V a
fe
½BaT½g½Ba
fe dV
1 2 Z
V s
fesT
½BsTfgg½Bs
fes dV
(4) and
R¼ Z
A
deT½NTfsgdA þ
Z
A a
fEagTqeðx; yÞdA (5)
¼ Z
A
deT
N
½ TfsgdA
Z
A a
feT
NpaT
qeðx; yÞdA: (6) Element stiffness matrix can be written as
R.L Wankhade, K.M Bajoria / Journal of Science: Advanced Materials and Devices 2 (2017) 255e262 256
Trang 3de
þ
Kse
de
¼
F1e
þ
Face
(7)
in which ½Ke ¼
Kde
þ
Kdae 1
Kaae
Kade
þ
Kdse
Ksse1
Ksde (8)
Faca
¼
Kdae
Kaae1
Qas
(9) where
Kde
¼
Z
V
B
½ T½ BC½ dV;Keda
¼
Kade T¼
Z
Va
B
½ T½ ½Be adV
Kaae
¼
Z
Va
½BaT½ ½Bg adV;
Kdse
¼
KsdeT
¼
Z
Vs
B
½ T½ ½Be sdV
and
Ksse
¼
Z
Vs
½BsT½ ½Bg sdV:
(10)
3 Finite elemnt modeling
3.1 Displacementfield
Fig 1shows the models under investigation for a
piezolami-nated beam with embedded and/or surface mounted piezoelectric
patches Displacementfield for the beam is considered as ‘u’ and ‘w’
along the x and z directions, respectively The higher order shear
deformation theory considering the effect of shear deformation is
adopted in the displacementfield
Hence, the displacementfield can be written as
u¼ u0þ zqxþ z2u*0þ z3q*x
where, u and w are the displacement of any point in the plate domain in lateral and transverse directions respectively u0and w0
are the displacement of midpoint of normal.qx,qyare the rotations
of normal at the middle plane in the x and y directions about the
y and x axes, respectively u*0; v*
0w*0;q*x and q*y are higher order terms
3.2 Strain within the element Strains associated with the displacementfield can be written as follows:
εp
|{z}
31
¼ εLþ εN
Strains are related to displacements using a strain displacement matrix as
ε
3.3 Electro-mechanical coupling For piezolaminated plates, two constitutive relationships exist which include the effect of mechanical and electrical loadings as given by eq.(5) Temperature variation effect is neglected in the formulation
R.L Wankhade, K.M Bajoria / Journal of Science: Advanced Materials and Devices 2 (2017) 255e262 257
Trang 4fDg ¼ ½efεg þ ½gfEpg
fsg ¼ ½Cfεg ½etfEpg (14)
8
>
>
<
>
>
:
sx 0
sy 0
sz 0
sx 0 y 0
sx 0 z 0
sy 0 z 0
9
>
>
=
>
>
;
¼
2
6
6
6
4
Q10 1 0 Q10 2 0 Q10 3 0 Q10 4 0 0 0
Q20 1 0 Q20 2 0 Q20 3 0 Q20 4 0 0 0
Q30 1 0 Q30 2 0 Q30 3 0 Q30 4 0 0 0
Q10 4 0 Q20 4 0 Q30 4 0 Q40 4 0 0 0
0 0 0 0 Q50 5 0 Q50 6 0
0 0 0 0 Q60 5 0 Q60 6 0
3 7 7 7 5
8
>
>
<
>
>
:
εx 0
εy 0
εz 0
εx 0 y 0
εx 0 z 0
εy 0 z 0
9
>
>
=
>
>
;
2
6
6
6
4
0 0 ez1
0 0 ez2
0 0 ez3
0 ey5 0
ex6 0 0
3 7 7 7 5
8
>
>
Exp0
Epy0
Epz0
9
>
>
Hence extension actuation mechanism
8
<
:
Dx 0
Dy 0
Dz 0
9
=
;¼
2
4 0 0 0 0 0 e
0 x6
0 0 0 0 e0y5 0
e0z1 e0z2 e0z1 0 0 0
3 5
8
>
>
<
>
>
:
εx 0
εy 0
εz 0
εx 0 y 0
εx 0 z 0
εy 0 z 0
9
>
>
=
>
>
; þ
2
4gx00x0 gy00 y 0 00
0 0 gz 0 z 0
3 5
8
>
>
Epx0
Eyp0
Epz0
9
>
For the piezoelectric layer polarized in the horizontal direction
i e parallel to the x axis, the dielectric displacement vector using
a direct piezoelectric equation is given as follows:
Hence for the shear actuation mechanism
8
<
:
Dx 0
Dy0
Dz0
9
=
;¼
2
4e
0
x1 e0x2 e0x3 0 0 0
0 0 0 0 0 e0y6
0 0 0 e0z5 0 0
3 5
8
>
>
<
>
>
:
εx 0
εy 0
εz 0
εx 0 y 0
εx 0 z 0
εy 0 z 0
9
>
>
=
>
>
; þ
2
4gx00x0 gy00 y 0 00
0 0 gz 0 z 0
3 5
8
>
>
Epx0
Eyp0
Epz0
9
>
where {D} is the electric displacement vector, [e] is the dielectric
permittivity matrix,ε is the strain vector, {g} is the dielectric matrix
{E} is the electricfield vector, [s] is the stress vector and [C] is the
elastic matrix for a constant electricfield
3.4 Electrical potential function
One electrical degree of freedom is used to consider
piezoelec-tric response Both actuator and sensor layers are separately
considered in the formulation Hence f0aand f0s are the electric
displacement at any point in the actuator and sensor layers,
respectively, the electrical potential functions in terms of the nodal
potential vector are written as
f0a¼
Npa
fe
f0s¼
Nps
where
Npa
and
Nps are the shape function matrices for the actuator and sensor layers, respectively.ffe
ag and ffeg are the nodal electric potential vectors for the actuator and sensor layers,
respectively
Micro- and nano-mechanical systems have dimensions which range in length from 1mm to 1000mm Such M/NEMS have thick-nesses typically in the range of few micrometers down to 25 nm or even sub nm regime which are bonded with the host materials As a result, the physical properties of the piezolaminated beams such as mechanical, electrical, thermal and magnetic properties can be different from the bulk values This of course depends on limitation and an opportunity as we can use these micro/nanostructures, such
as nanowires and nanobeams as the excellent systems for studying thickness effects in material properties and behavior at the small scale Among the different properties that arise when varying the size of any device we canfind changes in mechanical properties of the structure The latter case includes an anomalous behavior of their static and dynamic responses, the appearance of nonlinear damping, and the variation of the Young's modulus
4 Result and discussion 4.1 Smart beam with extension and shear piezo actuators The analysis of a piezolaminated beam with thickness-shear and extension piezoelectric actuators is performed The models con-structed are based on the higher-order shear deformation theory using eight node isoparametric elements Numerical examples of beams having piezoelectric actuators with different boundary conditions are presented The validity of the proposed models using extension as well as shear-mode actuators in smart beams is done
by comparing the results available in literature[4] The significance
of using thefinite element model adopted in the present work is illustrated to show its efficacy and efficiency in terms of the anal-ysis Further this adopted model can be effectively used to analyze beams with different boundary conditions and piezopatches with different shapes/sizes which are not possible by using the direct approach as reported in literature The models considered under investigation for the piezolaminated beam with extension and shear actuator mechanisms are shown inFig 1
In case of the extension mode, a three-layer piezolaminated beam with stacking sequence PZT5/Al/PZT5 is analyzed The beam
is 100 mm in length, whereas the thickness of an aluminum core is
16 mm and that of the piezo layer is 1 mm The PZT layers attached
to the top and bottom of the plate acts as actuators in the extension mode The PZT actuators are positioned with an identical poling direction along the thickness and are excited by a direct current voltage of 10 V The deflection induced by the actuators is calcu-lated for beams with different boundary conditions including cantilever, simply supported and clamped-hinged Dimensions of the beam are shown inTable 1
Furthermore, in case of the shear mode a three-layer lami-nated beam with stacking sequence Al/PZT5/Al is analyzed The length of the beam is taken as 100 mm whereas the thickness of the aluminum core is 8 mm and that of the piezo layer is 2 mm These dimensions are taken to make a comparison with the extension mode The PZT layer acts as an actuator and operates in the shear mode Here the actuator is excited by a direct current voltage of 20 V The response of beams with varying boundary conditions is presented inFigs 2e5 Material properties of the beams are obtained:
E¼ 70:3 GPa; m¼ 0:35 For PZT 5:
C11¼ 126 GPa; C12¼ 79:5 GPa; G12¼ G13¼ G23¼ 24:8 GPa;
m¼ 0:29;r¼ 7600 kg
m3; d31¼ d32¼ 166 pm=V;
R.L Wankhade, K.M Bajoria / Journal of Science: Advanced Materials and Devices 2 (2017) 255e262 258
Trang 5The shear deformation theory employed to model the
piezola-minated smart beams with extension and shear mode piezoelectric
actuators Exact solutions for these beams with various boundary
conditions are developed[4] Results obtained in the present work
are in good tune with the exact solution Comparison of the
transverse deflection of the cantilever beam with the extension and
shear modes is done showing that the tip displacement of the beam
within the extension mode is 74% higher than that of the shear
mode Thus the shear mode gives less deflection than that of the
extension mode Further, the piezolaminated beams with simply
supported, clamped, clamped-hinged features are analyzed
sub-jected to the piezoelectric effect
4.2 Control of the piezolaminated beam with an optimal location of actuator
The piezolaminated beam made up of the aluminum core having
a pair of linear actuators placed on either side is considered in the analysis to illustrate the optimization procedures with different boundary conditions The vibration equation of the beam is solved using the proposedfinite element method Thickness of the beam is taken as 1 mm, whereas the thickness of the piezoelectric actuator is taken as 0.4 mm The piezoelectric patch can be applied to each element The thickness of the bonding layer is not considered for simplicity The optimal location obtained from the procedure is used
Table 1
Dimensions of the smart beam with extension and shear piezoelectric actuators.
Smart beam with extension piezo actuator Smart beam with extension piezo actuator Stacking sequence PZT5/Al/PZT5 Al/PZT5/Al
Thikness of Aluminum core (t Al ) 16 mm 8 mm
Thickness of Piezoelectric layer (t p ) 1 mm 2 mm
Fig 2 Transverse deflection of the piezolaminated cantilever beam within the extension and shear modes.
Fig 3 Transverse deflection of the simply supported piezolaminated beam within the extension mode.
R.L Wankhade, K.M Bajoria / Journal of Science: Advanced Materials and Devices 2 (2017) 255e262 259
Trang 6tofind the optimal shape using the finite element procedure
Canti-lever andfixedefixed boundary conditions are used in the analysis
The properties of the aluminum beam are summarized inTable 2
The best location of one piezoelectric actuator is found using
the LQR based algorithm to be location 6 The dynamic simulation
of the beam with a uniformly distributed load of 5 sint N/m
applied through out the beam with a voltage of 100 sint V applied
to the actuators placed at the optimal location is done for 10 s
using the MATLAB function An optimal feedback controller is
designed using the gain obtained from MATLAB's LQR function
Fig 6shows thefirst four modes for the controlled response of the cantilever piezolaminated beam when the actuator is placed
at an optimal location
PZT is a chemically inert material which exhibits a high sensi-tivity of about 3mV/Pa and a large range of linearity up to an electric field of 2 kV/cm Hence it is possible to account for the fast response and long term stability high energy conversion efficiency, as observed inFig 6,of the dynamic response to control vibrations of the beams using PZT The maximum deflection is observed at the free end of the beam It is observed that in the second mode, the
deflection at the free end increased by 4.61% as compared to the first mode While in the third and fourth modes the deflection at the free end decreased by 3.22% as compared to thefirst mode Further, the piezolaminated beam having a piezoelectric patch
at the same optimal location is also examined in vibration In this case, the boundary condition of the beam isfixed at both the ends
Fig 7shows thefirst four modes for the controlled response of the
Fig 4 Transverse deflection of the clamped piezolaminated beam within the shear mode.
Fig 5 Transverse deflections of the clamped-hinged piezolaminated beam within the extension mode and shear modes.
Table 2
Material properties of the aluminum beam.
Property Aluminum PZT
Young's modulus GPA 79 63
Density kg/m 3 2500 7600
Piezoelectric constant m/V e 254*10 12
R.L Wankhade, K.M Bajoria / Journal of Science: Advanced Materials and Devices 2 (2017) 255e262 260
Trang 7Fig 6 Vibration modes of the cantilever piezolaminated beam.
R.L Wankhade, K.M Bajoria / Journal of Science: Advanced Materials and Devices 2 (2017) 255e262 261
Trang 8clamped piezolaminated beam when the actuator is placed at the
optimal location
About 72% reduction in displacement is observed for the
clam-ped beam as that of the cantilever beam for the same location of the
piezoelectric patch In case of the clamped piezolaminated beam,
the deflection in the second and fourth modes is increased by 5.5%
compared to thefirst mode While in the third mode, there is no
considerable change in the maximum deflection of the beam as
compared to thefirst mode Considerable deflection is controlled
by using the piezoelectric patch on the clamped beam as that of the
cantilever beam The smart beam with different boundary
condi-tions is studied to control response in vibration Cantilever and
clamped boundary conditions are considered for the analysis
Response in vibration for the cantilever beam is much faster than
for the clamped beam The best location of one piezoelectric
actuator is found using the LQR based algorithm to be location 6 for
which the response is studied in the analysis
5 Conclusion
Static and dynamic excitations are studied for the
piezolami-nated beams considering the numerical optimization technique
The piezolaminated beams are effectively used for shape and
vi-bration control in micro- and nano-mechanical systems due to their
sensing and actuation properties In the present work, the shape
control and vibration characteristics of the piezolaminated beams
are studied considering the extension and shear actuation
mech-anisms The shear deformation theory is employed to model the
piezolaminated smart beams with extension and shear mode
piezoelectric actuators The obtained results are in good agreement
with the exact solutions available in the literature A comparative
analysis of the transverse deflection of the cantilever beam in the
extension and shear modes shows that the maximum deflection for
the cantilever beam in the extension mode is 74% higher than that
in the shear mode Thus the shear mode gives less deflection than
that of the extension mode The piezolaminated beams with simply
supported, clamped, clamped-hinged features are analyzed
sub-jected to the piezoelectric effect Analysis of the vibration
charac-teristics of the piezolaminated beams is useful in identifying the
response of beams under static and dynamic excitations The
optimal location of the piezoelectric patch can be easily identified
for the corresponding boundary condition of the beam This work
can be further extended to verify the stability problem of beams with respect to piezoelectric actuation and sensing mechanism References
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