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Accepted Manuscript
Experimental and Mathematical Analysis of a Piezoelectrically Actuated Multilayered Imperfect Microbeam Subjected to Applied Electric Potential
Meisam Moory-Shirbani, Hamid M. Sedighi, Hassen M. Ouakad, Fehmi Najar
PII:
DOI:
Reference:
S0263-8223(17)32296-1
https://doi.org/10.1016/j.compstruct.2017.10.062
COST 9037
To appear in:
Composite Structures
Received Date:
Revised Date:
Accepted Date:
24 July 2017
13 October 2017
21 October 2017
Please cite this article as: Moory-Shirbani, M., Sedighi, H.M., Ouakad, H.M., Najar, F., Experimental and
Mathematical Analysis of a Piezoelectrically Actuated Multilayered Imperfect Microbeam Subjected to Applied
Electric Potential, Composite Structures (2017), doi: https://doi.org/10.1016/j.compstruct.2017.10.062
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Experimental and Mathematical Analysis of a Piezoelectrically
Actuated Multilayered Imperfect Microbeam Subjected to Applied
Electric Potential
Meisam Moory-Shirbani1, Hamid M. Sedighi1*, Hassen M. Ouakad2*, Fehmi Najar3
1
Mechanical Engineering Department, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz
61357-43337, Iran.
2
Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Dhahran, 31261,
Kingdom of Saudi Arabia.
3
Applied Mechanics and Systems Research Laboratory Tunisia Polytechnic School University of Carthage B.P. 743
- 2078, La Marsa, TUNISIA, Email: fehmi.najar@ept.rnu.tn.
*
Corresponding author email: h.msedighi@scu.ac.ir.
*
Corresponding author email: houakad@kfupm.edu.sa.
Abstract.
In this paper we propose to fabricate a microcantilever beam actuator using thin layer PZT material. The
multilayered microbeam is initially curved due to fabrication process. The device is also modeled using a
the Euler-Bernoulli beam theory and takes into account the multilayer structure of the device. The
inclusion of these imperfection into the theoretical model generates geometric nonlinearities. Static and
dynamic experimental analysis is conducted to examine the physiomechanical behavior of the considered
microstructure. To this end, The model is first compared and validated with experimental findings for an
applied constant voltage. Very good agreement depicts the accuracy of the proposed mathematical model.
When time varying voltage is applied, the nonlinear differential equation is solved using a perturbation
technique. Again good agreement with experiments are found. Finally, a parametric study is carried out to
explore the influences of different parameters on the frequency response of the piezoelectrically actuated
imperfect microbeams.
1. Introduction
Piezoelectric actuators are widely used as sensors, actuators and recently implemented in vibration energy
harvesting devices. Thanks to their transduction mechanism very simple designs are needed to actuate
these devices. Their implementation at the microlevel is easily obtained, such as for MEMS technology.
Since 1996, Ishihara et al. reviewed several piezoelectric microdevices used essentially as actuators [1].
For example Higuchi et al. [2] introduced a driving micromechanism using PZT layers for microrobot
arms. They reached 2mm/s as maximum velocity. Also, Fukuda et al. [3] utilized the same design to
actuate micromobile robot in water. A more recent review by Tadigadapa and Mateti [4] on piezoelectric
MEMS sensors reported several successful commercialized microdevices such as acoustic resonators, in
inertial sensors, high-frequency oscillators and filters, microactuators for RF applications, chip-scale
chemical analysis systems, etc.
Several researchers developed analytical models to analyze the influence of geometrical and physical
parameters on the general performance of mechanical systems [5-19]. DeVoe and Pisano [20] developed
a model describing the static response of ZnO piezoelectric cantilever beam. The experimental validations
of the actuator demonstrated the utility to optimize the layers position and thicknesses. A more detailed
1
model of multilayered beams is presented by Wolf et al. [21], in this model geometric nonlinearities due
to relatively large displacements are also taken into account. Mahmoodi et al. [22] also take into account
geometric nonlinearities for the equation of motion of a cantilever beam with a piezoelectric PZT patch.
A clear softening behavior was observed for large displacements. The results were confirmed by
experiments.
Several devices were proposed for microscale applications where layered structures are preferred because
of microfabrication restrictions. Piazza et al. [23] designed a microresonator by using a micromachined
clamped-clamped beam which includes a ZnO layer used for actuations. The design, tested
experimentally, showed very high mechanical quality factors with relatively low actuation voltages.
Gerfers et al. [24] modeled and fabricated a AlN piezoelectric microaccelerometers and reported high
signal-to-noise ratio and low dielectric loss, in addition to a good CMOS process compatibility. Thomas
et al. [25] reported a high level of parametric amplification for microresonator using PZT patches
deposited on a silicon clamped-clamped microbeam. The parametric amplification was observed up to a
factor of 14 in terms of the quality factor. Hofmann and Twiefel [26] designed a self-sensing bending
microactuator using a segmented electrodes. The proposed design was tested experimentally. Wang et al.
[27] presented a AlN-based inplane resonant microaccelerometer. Experiments showed a sensitivity as
high as 28.4 Hz/g which is 57% higher than previously reported data. Sarafraz and Seyed Roknizadeh
[28] studied the shape influence of a bimorph piezoelectric beam using COMSOL Multiphysics
software. Their results showed that due to the mechanical properties of the beams, the natural frequency
of the triangular beam is higher than the others for all considered parameters. Cheraghbak and Loghman
[29] investigated the magnetic field effects on the elastic response of polymeric piezoelectric cylinder
reinforced with carbon nanotubes. Theconsidered cylinder was subjected to internal pressure, a constant
electric potential difference at the inner and outer surfaces, thermal and magnetic fields. They indicated
that considering magnetic field can reduce the stresses of nano-composite cylinders.
The above mentioned devices are generally based on a multilayered structure having one or more active
layers. Their geometric configurations and electrical connections are essential to ensure optimal working
conditions. Zhu et al. [30] studied parallel and series connections for bimorph sensor with two
piezoelectric layers. They found that adding a third layer reduces the performance of the device. Also
within this context Jemai et al. [31] analyzed a multilayered cantilever beam with different number of
PZT piezoelectric layers and connection configurations for energy harvesting applications, they deduced
that an optimal number of layer should be used to increase the harvested energy.
Using microfabrication techniques, interdigitated electrodes can be deposited on a piezoelectric layers,
activating both the
d31 and d33 modes. Using these electrodes, Michael and Kwok [32] designed a micro-
lens actuator. For energy harvesting applications, Muralt et al. [33] demonstrated that the use of such
electrodes can achieve high voltage generation. Using also similar electrode design Jemai et al.
demonstrated that very good electromechanical coupling coefficient can be obtained if the geometrical
parameter of the interdigitated electrode design are correctly tuned [34, 35].
Modeling and design of these devices needs accurate material properties to be correctly identified
especially for thin film configuration. Since 1969, Holland and EerNisse [36] used a resonanceantiresonance method to determine piezoelectric coefficients. They reported several measurement
problems such as nonlinear behavior, material inhomogeneity or frequency dependence coefficient. They
proposed the measurement of the complex admittance at resonance and antiresonance to preserve phase
information and eliminate most errors. Lately, Liu et al. reviewed several measurement techniques for
2
piezoelectric thin films [37]. Direct and indirect methods were reported for 1.0 to 100.0 µ m film
thickness. They concluded that high resolutions in terms of displacement and charge are required to
achieve acceptable measurements. Lefki and Dormans [38] measured the piezoelectric coefficients of
PZT thin film using the normal load method with various compositions prepared by the sol-gel technique
or by organometallic chemical vapor deposition. Ren et al. [39] used also the normal load method to
measure piezoelectric coefficients of PZT thin films, but they proposed a composite tip to reduce the
damage caused by the metallic tip. Using a different measurement technique, Kholkin et al. [40] deduced
that a minimum of two microbeams should be used to obtain high-resolution interferometric
measurements of displacements in piezoelectric thin films.
Using quasi-static measurement, Dubois and Muralt [41] measured the effective piezoelectric transverse
coefficient
e31
for PZT and AlN using piezoelectric actuator to bend a cantilever microbeam with the
piezoelectric layer to be characterized. Wooldridge et al. [42] used a vertical comb drive MEMS actuator
to measure direct piezoelectric effect in PZT thin/thick films. [43] measured the PZT effective transverse
piezoelectric coefficient on anisotropic substrate using a cantilever microbeam. Guo et al. used a mini
impact hammer to measure
d33
coefficient of a PZT thin film [44]. A noncontact technique using
interferometry is used by Malakooti and Sodano [45] to determine the the
d33 and d31 piezoelectric strain
coefficients. Fialka and Beneš [46] compared several techniques for measurement of piezoeletric
coefficients. They used the frequency method, the laser interferometer technique, and the quasi-static
method. They concluded that the use of the frequency method is advantageous when we need to establish
individual coefficients or the complete matrix of the piezoelectric material coefficients. Tsujiura et al.
[47] compared direct and converse piezoelectric effects for the measurement of
e31,f coefficient for PZT
thin film using a unimorph cantilever microbeam. Polycrystalline and epitaxial PZT thin films were used.
They deduced that the polycrystalline PZT thin films show a clear nonlinear piezoelectric contribution,
however, the epitaxial PZT thin films depicted linear response.
Multilayered microbeams are generally subject to several additional difficulties in terms of modeling
because of their composite composition. In fact, multilayered microbeams are not symmetric in the
transverse direction, which deviate its neutral axis to non-symmetric positions [18]. In addition,
microfabrication techniques induce different residual stresses in the layers of the microbeam. This can
lead to an inevitable initial curling or deflection [48]: these cases are generally referred to as imperfect
microbeam microstructures. Initial deflection has been modeled by an initial axial force for piezoelectric
microbeam by Weinberg [49]. He deduced that using a linear model one can deduce the stiffness variation
due to this axial force. More recently, imperfect microbeams have been modeled using a bending
deflection defined at the initial unforced state of the structure [50]. This approach is simple to implement
and particularly valid for small curling of the beam. It has been also implemented for more complex
composite materials such as functionally graded microbeams [51]. Barati and Zenkour [52] proposed a
size-dependent nonlinear higher order refined beam model for post-buckling analysis of imperfect multiphase nanocrystalline nanobeams based on the modified couple stress theory. They employed a
micromechanical model based on Mori-Tanaka scheme to include the size of nanograins/nanopores. In
another research [53], they analyzed the post-buckling behavior of porous metal-foam nanobeams based
on a refined nonlocal shear deformation theory by considering an initial imperfection. They found that the
3
porosities have a significant effect on the post-buckling configuration of imperfect nanobeams. For large
initial bending deflections, curved beam and plate theories should be used [54].
In this paper we propose to fabricate an imperfect MEMS cantilever microbeam actuator using a PZT
layer. The device is modeled using a the Euler-Bernoulli beam theory and takes into account the
multilayer structure of the device. Due to fabrication residual stresses the microbeam is initially curved.
The inclusion of these imperfection into the theoretical model generates geometric nonlinearities. Static
and dynamic analyses from emprical and mathematical simulations well demonstrate the integrity of
proposed mathematical modeling to capture the mechanical behavior of multileyered microbeams with a
piezoelectric layer and initial imperfection.
2. Fabrication
The cantilever beams shown in Fig. 1 were all fabricated using a multi-mask fabrication process. The
bottom electrode has been spotted before growing the PZT. The beams have been fabricated in a 3 µm
silicon-on-insulator (SOI) wafer with 0.5 µm buried oxide (BOX) layer. During the first step, a 670 nm
silicon-oxide layer was grown. Then, a 10/100 nm Ti/Pt has been sputtered and patterned using the first
mask. A (100)-dominant thin film (250 nm) PZT has been grown using PLD on LaNiO3 as a seed layer.
All the films were cured at 600 C with an oxygen pressure of 0.1 mbar. On top of PZT, a 100 nm thick Pt
has been sputtered. Using the second mask, the top Pt layer has been patterned, followed by sputtering a
PZT using a wet etchant (the third mask). Around each fabricated beams, an area has been opened by
reactive ion etching (RIE) of the SiO2/Si/SiO2 layer stack (fourth mask). Finally, using the fifth mask, the
devices were released by isotropic etching of silicon, while the silicon device layer was protected by
photoresist. As seen in Fig. 1, PZT step-coverage has isolated the top and bottom Pt layers to prevent any
shortcut. The crystalline structure of the PZT thin-films was measured using a Philips XPert X-ray
diffractometer (XRD). A typical XRD pattern of the optimized PZT thin-films grown on 4-inch
Pt/Ti/SiO2/Si wafers, using large-scale PLD, is given in Fig. 2. The films were prepared at 600 ◦C with an
oxygen pressure of 0.1mbar. The θ-2θ scan clearly indicates the growth of PZT thin-films with (100)preferred orientation and no pyrochlore phase is observed. On top of PZT, 100nm thick Pt has been
sputtered. Using the second mask, the top Pt layer has been patterned, followed by patterning PZT using a
wet etchant (the third mask). Around the devices, an area has been opened by reactive ion etching (RIE)
of the SiO2/Si/SiO2 layer stack (fourth mask). Finally, using the fifth mask, the devices were released by
isotropic etching of silicon, while the silicon device layer was protected by photoresist. In this fabrication
process, the bottom Pt layer was etched under the top Pt contact-pads to minimize the parasitic
capacitances. As seen in Fig. 3a, PZT step-coverage has isolated the top and bottom Pt layers to prevent
the shortcut.
3. Mathematical Modeling and formulation
3.1 Geometry of the Multilayered Cantilever Microbeam
A schematic representation of a cantilevered multilayer microbeam with piezoelectric layer is shown in
Figure 3. The microbeam has length L and width w and is composed of a Silicon layer covered with
two SiO2 supporting layers and a PZT piezoelectric layer covered with two platinum (Pt) layers on its top
4
and bottom surfaces. The piezoelectric layer is bonded throughout the microbeam length. The
piezoelectric layer can be simultaneously actuated by a combination of DC and AC voltages.
Stress gradient across the thickness in thin microstructures such as, such as cantilever beams, can cause an
initial deformation in their geometry and hence may alter their functionality and may also lead to their
failure through deforming them either upward or even downward and consequently sticking them to the
substrate. In fact, such stress gradient across the thickness of a thin cantilever composite microbeam,
initially designed to be straight (Fig. 4a), can be induced due to a thermal mismatch between the beam
material and the substrate or among the various layers of the beam itself, in the case of a composite beam,
during material deposition. This causes an initial curvature in cantilever microbeams also called curling
[55] in cantilever beams, Fig. 4b. Therefore, a curled microbeam is a stress free structure that is brought
to its deformed shape by a stress gradient across its thickness.
3.2 Basic concepts and governing equations
According to the assumption of Euler-Bernoulli beam theory, the displacement field of a deflected beam
can be expressed as:
∂w( x, t ) 

−z


u
 x
∂x

  
u = u y  = 
0
(1)

u  w( x, t ) + w ( x) 
0
 z 



in which, w (x ,t ) and w 0 ( x ) denote the displacement and initial imperfection in transversal direction
associated with zero initial stress, respectively. It is assumed that x is the coordinate along the
microbeam length and y , z are the distance of each point on the cross-section from the neutral axis.
Referring to equation (1), the strain field of microbeam are written as [56]:
∂ 2 (w )
ε x = −z
= −zw ,xx
(2)
∂x 2
ε y = ε z = ε yz = ε xz = ε xy = 0
The stress-strain relation for the elastic layers are given by:
σ x ,i = C xx ,i ε x ,i i = 1, 2,3, 4, 6
(3)
where s x , e x and C xx represent the stress, strain and Young’s modulus in the axial direction (i.e., xdirection), respectively and i denotes the number of layer. The constitutive equations for the piezoelectric
layer (fifth layer) incorporating the couple effects of electro-mechanical behaviors is represent as [57]:
s x ,5 = C xx ,5e x ,5 - e zx E z
(4)
(5)
D z ,5 = e zx ,5e x ,5 + h zz E z
where Dz , eij and hij are the electric displacement in the transverse direction, piezoelectric and dielectric
coefficient of the PZT piezoelectric layer. The material properties and geometrical dimensions of different
layers is presented in table 1.
Table1. Material properties and geometric dimensions of the multilayered piezoelectrically actuated cantilever microbeam
5
Assuming the uniform electric field E z throughout the constant PZT piezoelectric layer thickness h5 ,
then the electric field parallel to the poling direction E z , in terms of the applied DC voltage V DC across
the thickness of the PZT piezoelectric layer, can be expressed as [57]:
Ez = -
VDC
h5
(6)
The location of neutral axis from the lower edge of the multilayered beam bender is denoted by z 0 which
can be determined using the static equilibrium conditions for the resultant axial forces (in the presence of
an acting external bending moment M) as [58]:
6
åF
x ,i
=0
(7)
i =1
By considering equations (2) to (4), the following relation for the axial force in the ith layer can be
obtained:
zi
Fx ,i = −b
∫C
xx ,i
wxx zdz
(8)
zi −1
and thereby, using equations (7) and (8) yields:
6
zi
∑∫C
xx ,i
zdz = 0
(9)
i =1 z i −1
in which the lower and upper integration limits of integral for ith layer, i.e. z i −1 , z i , can be determined as
follows:
i −1
zi = z 0 −
∑h
(10)
j
j =1
i
z i −1 = z 0 −
∑h
(11)
j
j =1
where h j is the height of ith layer. After some mathematical computations, equation (9) is re-written as
follows:
6
6
∑
i −1
6
C xx ,i [ zi 2 − zi −12 ] = 0 ⇔
∑
i =1
C xx , i [( z 0 −
i =1
∑
i
h j ) 2 − ( z0 −
∑h ) ]= 0
j =1
2
j
(12)
j =1
and thereby, the location of neutral axis from the bottom surface of multilayered microbeam can be
obtained as:
6
2
∑
i =1
z0 =
6
i
C xx ,i hi
∑
hj −
∑C
j =1
6
2
h2
xx ,i i
i =1
∑C
(13)
h
xx , i i
i =1
In order to derive the governing equation of motion, Hamilton’s principle is utilized. To this end, it is
necessary to compute the strain and kinetic energies of each layer as well as the works done by external
forces. The elastic strain energy of the multilayered microbeam Ps are given by [59]:
zi
i =6
∑∫ ∫
1
Πs =
2
i =1 Ai zi −1
1
σ x ,i ε x , i dzdA −
2A
z5
∫∫D
5
z ,5
E z ,5 dzdA5
(14)
z4
Substituting equations (2) to (14) into (14) yields:
zi
i =6
∑ ∫ ∫ zσ
1
Πs =
2
x ,i
wxx dzdAi − Π E = −
i =1 Ai zi −1
1
M x wxx dAi − Π E
2A
∫
(15)
i
in which the electrostatic energy of the PZT layer P E and bending moment M x are expressed by the
following relations:
z5
1
ΠE = −
2A
∫∫D
e
z ,5 zx ,5
5 z4
VDC
dzdA5
h5
i = 6 zi
Mx =
∑∫σ
(16)
i =6 zi
x ,i
zdz = −
i =1 zi −1
∑∫C
z5
∫
2
xx ,i
i =1 zi −1
wxx z dz + ezx ,5
z4
VDC
zdz ⇔ M x = − Dwxx + M piezo
h5
(17)
in the above equations, the total flexural rigidity D and the piezoelectric bending moment of the
multilayered system M piezo due to the applied voltage on the piezoelectric layer are calculated by:
D=
b
3
6
∑C
xx ,i
( zi 3 − zi −13 )
(18)
i =1
e zxV DC b 2
( z5 − z 4 2 )
2 h5
M piezo =
(19)
The kinetic energy of the system Pk is also determined as:
i =6
Πk =
zi
∑∫ ∫
i =1 Ai zi −1
L
ρi (
∂w 2
∂w
) dzdAi = m( ) 2 dx
∂t
∂t
0
∫
(20)
where m denotes the mass per unit length of microbeam which is given by:
i =6
m=
∑ bρ h
(21)
i i
i =1
The work done by the axial force N x can be obtained using the following relation [59]:
L
1
ΠF =
(N x )wx2 dx
20
∫
(22)
7
in which N x is computed by:
z5
z5
z4
z4
Nx = ∫ σ x,5bdz = ∫ (−Cxx,5w, xx z − ezx Ez )bdz = Nelectrical + Nunsymmetrical
(23)
where, N electrical and N unsymmetrical represent the axial forces due to the applied electrostatic voltage and
asymmetric configuration of multilayered microbeam. By submitting equation (4) into equation (23), the
exact expressions of the N electrical and N unsymmetrical are obtained as follows [59]:
Nelectrical = be zx ,5 (V DC +V A C )
N unsymmetrical = −
(24-a)
1
C xx ,5 b ( z5 2 − z 4 2 ) w, xx = K C w, xx
2
(24-b)
where K C in equation (24-b) indicates the coupling stiffness caused by the asymmetric arrangement of
piezoelectric layer. According to the Hamilton’s principle we have:
t
∫ (δΠ
k
+ δΠ F − δΠ s )dt = 0
(25)
0
Referring to equations (15), (20) and (22) and substituting into equation (25), one can obtain the
governing equation and boundary conditions for transversal vibration of cantilevered microbeam as
follows:
∂ 2M x
∂ 2w
∂2w
+ ( N electrical + Nunsymmetrical ) 2 + m 2 = 0
2
∂x
∂x
∂t
∂M x
∂w
+ ( N electrical + N unsymmetrical ) x ) = 0
w = 0 or (
∂x
∂x
(26)
−
wx = 0
or
(27)
Mx = 0
(28)
Using the relation described in equation (17), the governing differential equation in terms of w ( x ,t ) can
be extracted:
Dw xxxx + (N electrical + N unsymmetrical )w xx + mw = 0
(29)
3.3 Static analysis due to DC electric potential
In order to examine the static behavior of the multilayered microbeam subject to DC electric potential
load, the temporal terms in the governing equation (29) are omitted and then the linear static differential
equation can be expressed as:
Dws, xxxx + Nx ws, xx = 0
(30)
It is assumed that the analytical solution of equation (30) can be described as:
(31)
ws ( x) = AeSx
By substituting the static response (31) into (30), one can obtain the following auxiliary equation:
N
(32)
S 4 + Ex S 2 = 0
D
with roots:
S1,2 = ± -
N Ex
, S 3,4 = 0
D
(33)
and finally, the general static solution can be expressed as:
ws ( x) = A1eS1x + A2e- S1x + A3 x + A4
8
(34)
where the different constants A1 to A 4 are determined using the boundary conditions which for the
considered cantilevered microbeam are given by:
w s (0) = 0
(35)
(36)
w s ,x (0) = 0
(37)
M x (L) = 0
[
(38)
∂M x
∂w x
+ N Ex
]
=0
∂x
∂x x = L
By considering the above boundary conditions in the solution of equation (34), one can obtained the
coefficients as:
M piezo (e- S1L - 1)
(39)
A1 =
DS12 (2 - e- S1L - eS1L )
(40)
M piezo (e S1L - 1)
A2 =
DS12 (2 - e - S1 L - e S1L )
(41)
M piezo (eS1L - e- S1L )
A3 =
DS1 (2 - e- S1L - eS1L )
(42)
M piezo
M piezo
A4 =
DS12
=
N Ex
3.3 Dynamic analysis due to AC electric potential
Owing to the initial geometric imperfection of microbeam in the z direction, the dynamic governing
equation accounting for the initial rise w 0 ( x ) and subjected to the harmonic electric potential
V (t ) =V AC sin(wt ) is re-written as follows:
d + Dwd , xxxx + ( Nelectrical + KC wd , xx )[wd , xx + w0, xx ] = 0 Þ
mw
d + Dwd , xxxx + KC [wd , xx2 + wd , xx w0, xx ] + bezx,5VAC cos(w Et )[wd , xx + w0, xx ] = 0
mw
with the corresponding dynamic boundary conditions at x=0, L as:
w d (0,t ) = 0
w d , x (0,t ) = 0
(43)
(44)
(45)
- Dwd , xx ( L, t) + M Piezo = 0
(46)
Dw d ,xx x (L,t ) + N electricalw d ,xx (L,t ) = 0
(47)
while
M piezo =
ezxVAC b 2
( z5 - z4 2 ) sin(w t )
2 h5
(48)
Nelectrical (t) = bezx,5VAC cos(wEt)
(49)
In order to facilitate the analysis and obtain a generalized parametric solution, the following
nondimensional parameters are defined as:
t* =
w
w
D
x
t , x* = , w* = d , w0* = 0
4
L
L
L
mL
(50)
Substituting the dimensionless parameters into equation (43), the nondimensional nonlinear equation of
motion becomes:
9
d* + wd*. xxxx + N * cos(w E*t * )[ w*d , xx + w0,* xx ] + l 2 [(w*d , xx )2 + w*d , xx .w0,* xx ] = 0
w
(51)
where new terms are defined as:
bezx ,5VAC L2
KC L2
mL4
*
(52)
,l =
, w E* = w E
,
N =
D
D
D
In order to analytically study the dynamic behavior of multilayer microbeam, the Bubnov-Galerkin
decomposition procedure is employed to reduce the order of partial differential equation of motion and
discretize the governing equation. Therefore, the transverse displacement field can be assumed by the
following approximate solution [60]:
(53)
w( x* , t* ) = f ( x* )h (t* )
where h (t * ) represents the generalized coordinate and f ( x * ) denotes the first eigenfunction for the
transverse motion of cantilevered beam which can be expressed as [58]:
cos a + cosh a
(sinh a x * - sin a x * )], a = 1.8751
f (x ) = [(cosh a x * - cos a x * ) (54)
sin a + sinh a
By applying the weighted residual Bubnov-Galerkin method [61], the nonlinear governing equation of
motion is reduced to:
(55)
h + wo2h = [ea2h 2 + e F1 cos(wE* t* )h + e 2 F0 cos(wE* t* )]
in which different parameters appeared in the equation (55) are defined as follows:
1
∫
ωo 2 =
[φ (x ∗ )
0
2
∗
2
∗
d 4φ (x ∗ )
∗ d φ (x )
∗
∗ d φ (x )
2
x
w
x
+
λ
φ
(
)
(
)
]dx ∗
0
dx ∗4
dx ∗2
dx ∗2
1
∫[(φ(x
∗
)φ (x ∗ ))dx ∗
0
1
∫
−λ 2 [φ (x ∗ )
0
εα 2 =
∗
d φ (x ) d 2φ (x ∗ ) ∗
]dx
dx ∗2
dx ∗2
2
1
∫[(φ(x
∗
)φ (x ∗ ))dx ∗
0
(56)
1
d 2φ ( x∗ ) ∗
− [(φ ( x∗ )( N ∗ w0∗ ( x∗ ))
]dx
dx∗2
0
∫
ε 2 F0 =
1
∫[(φ (x )φ(x ))dx
∗
∗
∗
0
1
∫
− [(φ ( x∗ )( N ∗ )
ε F1 =
0
d 2φ ( x∗ ) ∗
]dx
dx∗2
1
∫[(φ (x )φ (x ))dx
∗
∗
∗
0
3.4 The method of multiple scales
The idea of the multiple scales method is to express the solution of ordinary differential equations as a
function of multiple independent time-scales. In this procedure, the independent variables T n known as
time-scales are defined by [61]:
(57)
Tn = e nt * , n = 1,2,3,....
10
where ε << 1 is a small perturbation parameter and the derivatives with respect to nth time-scale is
expressed by the symbolic operator D n as follows:
∂
= Dn
∂Tn
(58)
By considering the above assumptions, the derivatives with respect to the dimensionless time t * can be
defined using the following expansions in terms of the partial derivatives with respect to time-scales T n :
d
= D0 + e D1 + e 2 D2
(59)
*
dt
(60)
d2
= D02 + 2e D0 D1 + e 2 ( D12 + 2 D0 D2 )
*2
dt
and then the approximate solution of equation (55) can be assumed by an expansion series as a function of
both t * and ε according to:
(61)
h (t* , e ) = h0 (T0 ,T1,T2 ) + eh1 (T0 ,T1 ,T2 ) + e 2h2 (T0 ,T1 ,T2 ) + O(e 2 )
Substituting (59) to (61) into (55) and equating the coefficients of ε n equal to zero, one gives the
following linear ordinary equations:
ε0 :
∂ 2η 0
+ ω02η0 = 0
∂T0 2
(62)
ε1 :
∂ 2η1
∂ 2η 0
+ ω 02η1 = F1 cos(ω0 T0 + σ T2 )η 0 + α 2η 02 − 2
2
∂T0 ∂T1
∂T0
(63)
ε2 :
∂ 2η 2
∂ 2η1
∂ 2η 0
2
+
ω
η
=
F
cos(
ω
T
+
σ
T
)
+
F
cos(
ω
T
+
σ
T
)
η
+
2
α
η
η
−
2
−
0 2
0
0 0
2
1
0 0
2
1
2 0 1
∂T0 ∂T1 ∂T12
∂T0 2
(64)
in which the dimensionless frequency of the AC voltage w E* is perturbed around the fundamental
frequency w0 in the following from:
(65)
wE* = w0 +e2s
where s denotes the detuning parameter. It is convenient to express the solution of the first equation (62)
in terms of complex function A (T1 ,T2 ) and its complex conjugate A (T1,T2 ) by the following form:
h0 = A(T1,T2 )eiw0 T0 + A(T1,T2 )e-iw0 T0
(66)
Substituting equation (66) into (63) gives:
∂ 2η1
F
∂A
+ ω02η1 = 1 Aei (2ω0 T0 +σ T2 ) + α 2 A2 e 2iω0 T0 − 2i
ω 0 eiω0 T0 + α 2 AA
2
∂T1
∂T0 2
(67)
By vanishing the secular terms of the right hand side of equation (66), leads the function A to become
independent of the time scale T1 . Consequently, one can obtain the particular solution for h 1 as:
h1 =
F1 Aei ( 2w 0 T0 +s T2 ) 2 F1 Ae - is T2 + 4a 2 AA a 2 A2 e 2iw 0 T0
+
6w 02
6w 02
3w 02
(68)
Substituting for h 0 and h 1 from equations (66) and (68) into equation (64) and eliminating the secular
terms in the right hand side of consequence equation leads to:
−2iω0
5α F AA F
F2A
5α F A2
10α 2 2 A2 A F12 A
∂A
+ [ 2 12 + 0 ]eiσ T2 + [ 1 2 ]e 2iσ T2 + [ 2 12 ]e − iσ T2 +
+
=0
∂T2
2
3ω 0
4ω 0
6ω 0
3ω02
6ω 02
(69)
To solve the above equation with respect to A (T1 ,T2 ) , it is suggested to assume A in the polar form as:
1
A = aeib (T2 )
2
(70)
11
in which a and β are real functions. By introducing new parameter γ = σT 2 − β , after some
complicated mathematical computations and separating the result into real and imaginary parts, for the
steady state solution we have:
a′ =
aγ ′ =
1
1
[12 F0ω02 + 5α 2 F1a 2 ]sin(γ ) +
[3F12 ]a sin(2γ ) = 0
24ω03
24ω03
(71)
F1
5α 2 2 3
1
1
2
2
2
[12
+
15
]cos(
)
+
[3
]
cos(2
)
+
[
+
]
+
[
]a = 0
F
ω
α
F
a
γ
F
a
γ
σ
a
0
0
2
1
1
24ω03
24ω03
12ω 03
12ω 03
(72)
where the prime stands for the derivative with respect to the second order time-scale T2 . By solving the
couple equations (71) and (72) for a and γ , and referring to equation (70), the approximate solution of
the nonlinear governing equation (55) using the method of multiple scale can be concluded as follows:
wd∗ ( x ∗ , t ∗ ) = [ a cos(ω E∗ t ∗ − γ ) −
ε F1 a
εα 2 a 2
εα 2 a 2 + ε F1 a cos(γ )
∗ ∗
∗ ∗
cos(2
t
)
cos(2
t
2
)
]φ ( x∗ )
ω
−
γ
−
ω
−
γ
+
E
E
6ω 02
6ω02
2ω 02
(73)
4. Results and Discussion
In this section, at first, the static and dynamic experimental analysis is conducted to verify the soundness
of mathematical modelling and empirically extract the fundamental frequency of multilayered microbeam
with initial geometric imperfection. Then, a parametric study is carried out to investigate the dynamic
behavior of piezoelectrically actuated microbeam.
4.1 Experimental analysis
Due to improper fabrication of microstructures in the manufacturing treatments, it is expected to create
microbeams with initial geometric imperfection which affects the physiomechanical behavior of such
systems. After some calibration methods, it was found that the considered multilayered microbeam
contains an initial curvature proportional to its static deformation, i.e. w0 ( x) = W0 ws ( x) . The static results
are reported in figure 5 for different lengths of microbeam. As evident from the presented findings, one
can found that the theoretical results are in good agreement with those acquired by experimental analysis.
Thus, it is concluded that the linear governing equation described in equation (30) as well as the analytical
solution provided by equation (34) could predict the static behavior of the system with acceptable
accuracy. It should be pointed out that the initial imperfection of the microbeam is shown with blue
curves.
To verify the accuracy of nonlinear modeling of vibrational equation of motion (55) and also the
analytical approximation provided by (73) to capture the dynamic behavior and fundamental frequency of
cantilevered multilayer microbeam, the theoretical results together with the experimental measurements
are plotted in figure 6 for some specific microbeam's lengths. One can infer that the simulated dynamic
results are in consistent with those of experimental findings as shown in figure 3.
In addition, in Table 2, we especially compared the fundamental frequency obtained from theoretical and
experimental analyses for different lengths. As can be seen, the maximum deviation is less than 1% which
is worthy of acceptance. It is demonstrated that our proposed theoretical models can be genuinely
employed in the design of imperfect multilayered microbeams in order to estimate their fundamental
natural frequencies.
Table 2. A comparison between fundamental frequencies of the multilayered microbeam obtained by theory and
experimental methods
Beam Length
L=300µm
L=210µm
Measurement (kHz)
39.49
72.19
12
Theory (kHz)
39.20
72.15
Relative error %
0.74
0.06
L=170µm
L=140µm
L=120µm
L=100µm
102.31
140.86
178.63
232.14
101.98
139.77
178.41
230.70
0.33
0.78
0.12
0.63
4.2 Parametric analysis
To examine the performance of cantilevered multilayer microbeams as an energy harvester device, it is
necessary to capture the vibrational characteristics and frequency response of the structure. To this end,
the natural frequencies of the considered multilayer microbeam as function of its length for different
values of the thickness ratio h5 / ht are presented in figure 7. It is exhibited that the fundamental frequency
have a descending trend with respect to the beam length. On the other hand, it is shown that any increase
in the thickness ratio leads to gradual increase in the fundamental frequency. As the beam length
increases, the sensitivity of frequency to this dimensionless parameter is decreased.
Moreover, figure 8 depicts the variations of the fundamental natural frequency versus the thickness ration
for some assigned values of Young’s modulus of piezoelectric layer C x x ,5 . One can observe that any
increase in the Young’s modulus results in a significant increase in the natural frequency of the
multilayered microbeam. In addition, it is evident that there is a linear relationship between the natural
frequency of the system and the thickness ratio h5 / ht .
Figures 9 and 10 indicate the influence of excitation amplitude V A C and thickness ratio h5 / ht on the
frequency response of the considered system. One concludes that as the amplitude of excitation increases
the microbeam shows the higher amplitude of vibration. Moreover, according to the simulated results in
figure 10, it is evident that decreasing the thickness ratio leads to the higher amplitude of vibration and
therefore increase the efficiency of the system as an energy harvester.
5. Conclusion
We designed and fabricated a microcantilever beam actuator using thin layer PZT material. The device is
composed of 6 layers formed by the substrate materials, active layer and the electrodes. Due to fabrication
residual stresses and dissymmetry of the microbeam, it was found initially curved. The device was
modeled using the Euler-Bernoulli beam theory. The model takes into account the multilayer structure of
the device. When taking into account the initial curvature of the microbeam into the proposed theoretical
model, it generates geometric nonlinear terms in the equation of motion.
The model was compared and validated with experimental findings for a constant applied voltage. Very
good agreement are fond between the results depicting the accuracy of the proposed mathematical model.
When time varying voltage is applied, the equation of motion and associated boundary conditions are
discretized using an assumed mode approach. Therefore, a perturbation technique is applied to generate
the frequency-response curves. Again good agreements are found with experiments.
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Fig. 1. Scanning electron micrograph (SEM) of a PZT-on-silicon cantilever resonator
Fig. 2. The XRD pattern of PZT thin-films on the Pt/Ti/SiO2/Si wafers
18
Fig. 3. a) The top-view and b) The cross-section schematics of the device
Curling Process
(a)
(b)
Fig. 4. Composite cantilever beam deforms due to stress gradient across the thickness
19
(a)
20
(b)
21
(c)
22
(d)
23
(e)
24
(f)
Fig. 5. Comparison of analytical and experimental results of the static transverse deflection for a) L=100µm b) L=120µm c)
L=140µm d) L=170µm e) L=210µm f) L=300µm
25
(a)
26
(b)
27
(c)
28
(d)
29
(e)
30
(f)
Fig. 6. Comparison of analytical and experimental results of the dynamic transverse deflection with applied AC frequency for a)
L=100µm b) L=120µm c) L=140µm d) L=170µm e) L=210µm f) L=300µm
31
Fig.7. Variation of natural frequency calculated using the theory model with beam length L for different values of h5 / ht
32
Fig. 8. Variation of natural frequency calculated using the theory model with thickness ratio h5 / ht for different values of C xx ,5
33
Fig. 9. Variation of maximum dynamic transverse deflection calculated using the theory model with applied AC frequency for
different values of VAC and for when L=300µm
34
Fig. 10. Variation of maximum dynamic transverse deflection calculated using the theory model with applied AC frequency for
different values of h5 / ht
35
36
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