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Forced transverse vibration analysis of a circular viscoelastic polymeric piezoelectric microplate with fluid interaction.

1. Introduction

Introducing smart materials in MEMS application for sensing and actuation is of interest to many researchers. Some materials such as piezo-ceramics are used in many micro-devices. However, they exhibit some limitations in their use. Polymer piezoelectric materials are one of the substituting materials which can pass these limitations.

Kawai (1969) investigated the piezoelectricity of polymers on poly (vinylidene fluoride). Piezoelectricity in ferroelectric polymers is surveyed by Furukawa (1984). A review on the piezoelectric polymers has been made by Fukada (2000). Riande and Calleja (2004) studied the electrical properties of polymers. Tanaka, Tanaka, and Chonan (2008) used PVDF films for evaluating tactile sensing. Carpi and Smela (2009) perused biomedical application of electro-active polymers. Seminara et al. (2011) set up an experimental procedure for measuring the PVDF piezoelectric characteristics. Chiu et al. (2013) developed a PVDF sensor patch for heartbeat monitoring. No theoretical modelling has been mentioned in this article. Moleiro, Mota Soares, and Mota Soares (2014) proposed an exact solution for static analysis of multi-layered PVDF. Different loading conditions and plate aspect ratios were investigated. Jaitanong et al. (2014) examined piezoelectric properties of cement-based/PVDF/PZT. Duan et al. (2015) used PVDF piezoelectric films as sound absorption. He presented a flexible micro-perforated panel based on PVDF films for this purpose. Among these literature reviews, no vibration analysis of monolayer PVDF microplate as an actuator is reported. Monemian Esfahani and Bahrami (2016) investigated free vibration analysis of PVDF circular microplate. In this article, viscoelastic properties of polymer are not considered.

In addition to these studies, some researchers proposed numerical analysis of the piezoelectricity to predict the behaviour of these materials. Tzou and Tseng (1990) used finite element method to solve a two-dimensional piezoelectric material for an actuator or a sensor. Kagawa, Tsuchiya, and Kataoka (1996) used FEM for simulating a piezoelectric actuator. Khutoryansky, Sosa, and Zu (1998) applied boundary element method (BEM) to model and simulate active materials. Benjeddou (2000) studied FEM for some structures such as shells and plates. Jun and Zhaowei (2002) surveyed FEM to investigate the performance of piezoelectric actuator in hard disk drives. Sharif-Khodaei and Aliabadi (2014) used delay delay-and-sum algorithms for damage detection using piezoelectric patches. Guanghui Qing, Qiu, and Liu (2006) proposed a semi-analytical solution for plates with piezoelectric patches. Zou, Benedetti, and Aliabadi (2014) used boundary elements method for piezoelectric transducers for health monitoring of structures.

In this paper, governing equations are derived using piezoelectric constitutive equations in fully general form. Fluidic media is modelled as a damping foundation acting under the plate. Viscoelastic properties of PVDF are introduced to the governing equations using Kelvin-Voigt laws. Boundary conditions are assumed clamped at all edges, which is a condition for most micro-devices although it adds more complexity to the problem. Since the governing equation consists of temporal and spatial parts, it needs two methods to solve it. At first, assumed mode method is used for spatial part of the equation. The output of this section is a matrix-type equation which only depends on time. The new equation is then solved using Newmark's [beta] method. Combining these two methods will result in the solution of the problem. MATLAB codes are generated to solve the equation. Finite element model is then developed using COMSOL Multiphysics package to verify the solution. The effect of parameters such as damping coefficient, viscoelastic parameter, input voltage and excitation frequency is investigated.

2. Problem formulation

Figure 1 shows the configuration of the microplate. The plate is clamped at outer edges and the relative applied voltage on top and bottom of the microplate is [V.sub.rel].

The piezoelectric constitutive equations (see Appendix 1) in matrix form in cylindrical coordinates for a piezoelectric material in the most general formula, considering thin plate theory where [[epsilon].sub.zz] is zero, can be rewritten as follows:

[mathematical expression not reproducible] (1)

[mathematical expression not reproducible] (2)

where r is the radius and [theta] is the angle in cylindrical coordinate. [sigma] is the stress on the plate and its superscripts denote the direction. [epsilon] is strain, E is electrical field vector, D is electrical displacement vector, [xi] is permittivity, V is applied voltage and [C.sub.pq.sup.E] = ([[partial derivative][[sigma].sub.p]/[partial derivative][[epsilon].sub.q]]) is elastic matrix. e is e = [xi]*h where h is piezoelectric constant.

The electric displacement components must satisfy Equation (2) (Ruan et al. 1999):

[mathematical expression not reproducible] (3)

Substituting Equations (4) into (5) and assuming [E.sub.z] = [[partial derivative]V/[partial derivative]z] where V is the applied electric potential result in:

[mathematical expression not reproducible] (4)

Based on Kirchhoff plate theory for cylindrical coordinates, see Appendix 2, the relation between applied electric potential (V) and transverse displacement (w) is as follows:

[mathematical expression not reproducible] (5)

If relative electric potential between top and bottom of the plate assumed [V.sub.rel], by integrating Equation (10) twice, the following equation will be obtained:

[mathematical expression not reproducible] (6)

For a viscoelastic material, which obeys the Kelvin-Voigt laws, Hooke's law is written as follows:

[mathematical expression not reproducible] (7)

where [eta] is the viscous coefficient. Over dot shows derivation with respect to time.

Expanding Hooke's law for linear material, the total stress in a viscoelastic piezoelectric plate is:

[mathematical expression not reproducible] (8)

[mathematical expression not reproducible] (9)

[mathematical expression not reproducible] (10)

Using these equations, the resultant bending moments are:

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

[mathematical expression not reproducible] (13)

and membrane inplane forces are as follows:

[mathematical expression not reproducible] (14)

[mathematical expression not reproducible] (15)

The resulting shear forces can be obtained from the following equations:

[mathematical expression not reproducible] (16)

[mathematical expression not reproducible] (17)

The dynamic equilibrium equation in z direction for an infinitesimal element of a circular plate is given as:

[mathematical expression not reproducible] (18)

where c is the damping coefficient of the fluid.

Substituting Equations (19)-(22) into (23), the governing equation is expressed as:

[mathematical expression not reproducible] (19)

3. Numerical simulation

Without loss of generality and for simplification, axisymmetric case is considered. Therefore, all derivatives with respect to [theta] are omitted. In order to solve Equation (24), assumed mode method is applied. Assume the spatial functions as follows:

w(r, t) = [N.summation over (i=1)] [T.sub.i](t)[[phi].sub.i](r) (20)

Here, [[phi].sub.i](r) is a mode shape function which satisfies the boundary conditions. Substituting Equation (25) into (24), multiplying both sides by [[phi].sub.j](r) and rewriting it in matrix form, one can obtain,

[mathematical expression not reproducible] (21)

[mathematical expression not reproducible] (22)

[mathematical expression not reproducible] (23)

[mathematical expression not reproducible] (24)

[mathematical expression not reproducible] (25)

From Equation (26), one can find that the solution of this equation with zero initial conditions is zero. Consequently, we need initial conditions or applying a pressure on the plate to deform it. Latter is chosen for the problem. By inserting a pressure in Equation (24) and continuing with the equations, Equation (26) will change into Equation (31),

[mathematical expression not reproducible] (26)


[mathematical expression not reproducible] (27)

4. Mode shapes generation

In order to apply this method, mode shape functions are needed. Since the plate is axisymmetric and clamped at the outer edge, the first mode shape is chosen as follows: (Chakraverty 2009),

[[phi].sub.1](r)) = [(1 - [[r.sup.2]/[R.sup.2]]).sup.2] (28)

Next shape functions are generated using recurrence scheme as follows:

[[phi].sub.2](r)= ([r.sup.2] - b)[circ.phi.sub.1](r) (29)

[[phi].sub.k+1](r) = ([r.sup.2] - [b.sub.k])[circ.phi.sub.k](r) - [c.sub.k][circ.phi.sub.k-1](r); k [greater than or equal to] 2 (30)


[mathematical expression not reproducible] (31)

[mathematical expression not reproducible] (32)


[circ.phi.sub.i] = [[phi.sub.i]/[a.sub.i]] (33)

[mathematical expression not reproducible] (34)

5. Numerical results

Newmark's [beta] method is used in order to solve Equation (31). Newmark (1959) proposed a method for solving problems in structural dynamics. This method starts with an initial condition and continues in each time step to find the next position, velocity and acceleration. The relation between each iteration is as follows:

[mathematical expression not reproducible] (35)

[mathematical expression not reproducible] (36)

where [beta] and [gamma] are the method's parameters.

The incremental form of the differential equation is given as follows:

[mathematical expression not reproducible] (37)

Substituting Equations (36) and (37) into (38) and solving for [{T}.sub.n+1], the recurrence relations will be obtained.

To find the solution, MATLAB code is written and the appropriate results are achieved.

Table 1 shows the parameters used for numerical simulation.

6. Free vibration analysis

Free vibration analysis is solved to find natural frequencies of the microplate. For this purpose, the matrices C, q and P are assumed zero. Figure 2 shows frequency vs. maximum displacement.

7. Finite element modelling

In order to verify the results, a finite element model is developed using COMSOL Multiphysics package. For this purpose, a 2D axisymmetric model of the plate is built. The physic of piezoelectric devices and laminar flow were applied to the model. Time-dependent study was selected to find the microplate behaviour and compare the results with MATLAB code. Figure 3 shows meshing and geometry of the plate.

Figures 4-6 show the comparing results between COMSOL model and MATLAB code.

It can be seen that the results have an excellent agreement. It can be concluded that the governing equation and its solution can estimate the behaviour of the plate under the assumed conditions. It worth mentioning that the shift which appears in the plots from zero is due to the applied pressure. This pressure deforms the plate and then the plate vibrates because of the applied voltage.

8. Results

The following figures are obtained from MATLAB code using assumed mode and Newmark's [beta] method.

Figure 7 shows the effect of different input voltage on the amplitude of the microplate.

As expected, by increasing the input voltage, the amplitude is increased.

Figure 8 shows the changes in microplate vibration by changing the excitation frequency.

Figure 9 shows the effect of damping coefficient on the behaviour of the plate. Viscosity of the plate is not considered here.

By increasing the damping coefficient of the fluid, the amplitude of the vibration is decreased.

Figure 10 shows the effect of viscoelasticity of the microplate. In order to investigate this effect, it is assumed that the plate has no interaction with fluid and in fact the damping coefficient is zero.

The same behaviour occurred for the viscosity of the microplate. However, its effect is less than the damping coefficient.

Figure 11 shows the effect of changes in damping coefficient with viscoelastic microplate.

Figure 12 shows the effect with changes in the viscoelastic parameter while the plate interacts with the fluid.

Figure 13 shows the variation of damping parameter and input voltage.

By increasing the damping coefficient, the amplitude of the vibration is decreased and also the microplate will tend to its stable position more slowly. The stable position is where the applied pressure, discussed in section 3 and 7, deforms the plate.

9. Conclusion

In this paper, forced vibration analysis of a viscoelastic polymeric piezoelectric microplate was investigated. The microplate was assumed circular and clamped in the outer edge. The effect of viscoelasticity was modelled using Kelvin-Voigt laws. Interaction of the plate and the fluid was modelled as a damping foundation under the plate. The governing equation was then derived and was solved using assumed mode method and Newmark's [beta] Method. The results were compared to the developed finite model using COMSOL Multiphysics package. Comparing the results showed an excellent agreement between two methods which verified the governing equation and its solution. The effect of frequency, input voltage, damping coefficient and viscose parameter on the behaviour of the microplate was discussed. The results showed that input voltage and its maximum amplitude have a direct relation to the amplitude of the vibration while the damping coefficient and viscose parameter have an inverse relation with the amplitude.


This project is supported by IRAN National science foundation (INSF).

Notes on contributors

A. Monemian Esfahani was born in Esfahan, Iran and is currently a PhD candidate at the Deportment of Mechanical Engineering, Amirkabir University of Technology (AUT). He received his BSc and MSc degrees in Mechanical Engineering at AUT in 2007 and 2009, respectively. He served as an assistant professor for several years in the fields of automatic control, mechatronics and microsystems (MEMS and BioMEMS). His research interests include mechanical vibrations, design, automatic control, MEMS and BioMEMS. Email:

M. Bahrami received his BSc from Tehran Polytechnic in 1975, and his MSc in 1977 and PhD in 1981 from Oregon State University, all in Mechanical Engineering. He joined Amirkabir University of Technology (Tehran Polytechnic). In his endeavour at AUT, he developed several courses in Control Engineering, Robotics and MEMS. He established Robotics and Automation Laboratory and developed relations with industry. Later, he founded and directed the New Technology Research Center. He served as the general director of Research in Ministry of Science, Research and Technology. Bahrami's research interests are MEMS/NEMS, Robotics and Technology Foresight.

S. R. Ghaffarian Anbaran received his BSc in polymer engineering and MSc and PhD both in polymer engineering, from UMIST, UK in 1987 and 1992, respectively. He served as a professor at Amirkabir University of Technology (AUT) from 1992. He is the fellowship of several international associations. His research interests lie in the broad fields of polymer composite, including smart polymers, nano-composite and liquid crystal polymers.


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Appendix 1

The piezoelectric constitutive equations for actuation are as follows (Jalili 2010):

[[sigma].sub.p] = [c.sub.pq.sup.E][[epsilon].sub.q] - [e.sub.pi][E.sub.i] (1)

[D.sub.i] = [e.sub.ip][[epsilon].sub.q] + [[xi].sub.ij.sup.[epsilon]][E.sub.i] (2)

where [epsilon] is the strain vector, [sigma] is the stress vector, E is the electric field vector, D is the displacement vector, s is the compliance coefficients matrix, d is the matrix of piezoelectric strain constants, [xi] is the permittivity constants matrix, [c.sub.pq.sup.E][[epsilon].sub.q] = [([partial derivative][[sigma].sub.p]/[partial derivative][[epsilon].sub.p]).sub.D] is the elastic stiffness coefficients matrix under constant dielectric displacement and [beta] refers to the constant or zero strain condition for the impermittivity constants matrix. Indices i, j = 1, 2, 3 and p, q = 1, 2, ..., 6 refer to different directions within the material coordinate systems. [h.sub.pi] = -[1/[d.sub.pi]] is the matrix of piezoelectric constants and e = [xi]*h.

Appendix 2

All displacements can be related to transverse displacement by the following equations:

[[epsilon].sub.rr] = -z[[partial derivative.sup.2]w/[partial derivative][r.sup.2]] (1)

[mathematical expression not reproducible] (2)

[mathematical expression not reproducible] (3)

A. Monemian Esfahani (a), M. Bahrami (a) and S. R. Ghaffarian Anbarani (b)

(a) Department of Mechanical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran; (b) Department of Polymer and Color Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

CONTACT M. Bahrami


Received 14 February 2016

Accepted 6 February 2017
Table 1. Parameters.

Parameter           Value               Unit

E               1.27x[10.sup.9]           pa
v               0.225                     1
[rho]        1780                    kg/[m.sup.3]
h              10 x [10.sup.-6]           m
a             500 x [10.sup.-6]           M
[d.sub.31]     23 x [10.sup.-12]         m/V
[d.sub.32]    [23 x [10.sup.-12]/5]      m/V
[xi.sub.33]   106 x [10.sup.-12]         F/m
[eta]           2 x [10.sup.10]           s
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Author:Esfahani, A. Monemian; Bahrami, M.; Anbarani, S.R. Ghaffarian
Publication:Australian Journal of Mechanical Engineering
Geographic Code:7IRAN
Date:Mar 1, 2018
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