Nonlinear vibration analysis of partially coated circular

microplate under electrostatic actuation

R. Sepahvandi , M. Zamanian* , B. Firouzi, S. A. A. Hosseini

Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, P.O. Box 15719-14911,

Tehran, Iran

* Corresponding author: E-mail address: zamanian@khu.ac.ir , Tel: +98 263 4579600,

Abstract

This paper investigates the optimal configuration for a partially two-layered circular capacitive microplate

subjected to AC-DC electrostatic actuation. To this end, the static deflection due to DC electrostatic

actuation, natural frequency of vibration about static position and primary resonance response due to AC

electrostatic actuation are studied. Primarily, the nonlinear equations of motion are derived through classical

laminated plate theory (CLPT). Then, the static position and natural frequency of vibration around static

position are obtained using Galerkin approach. The forced vibration equations around static position are

separated using Galerkin method, and solved by the multiple scale perturbation theory. Firstly, the impact of

changes in the second layer radius on the variations of static and dynamic response of the system is studied

while its thickness remains constant. Then, the effect of changes in the second layer thickness is studied

while its radius remains constant. Finally, the impact of simultaneous change in the radius and thickness of

the second layer is studied while its volume remains constant. The results show that the highest frequency

and lowest static deflection occurs when the second layer covers fifty percent of the first layer. This result

can be used for designing high speed microsensors.

Keywords: Electrostatic actuation, Circular microplate, Static deflection, Primary resonance, Perturbation

theory

1. Introduction

Microelectronic integrated circuits can be as the thinking minds of systems, and microelectromechanical

systems develop this thinking power by addition of eye and arm to them, so that microsystems can sense and

control surrounding environment. The main element of many microelectromechanical systems such as

micropumps, microphones and microsensors is a circular microplate under the electrostatic actuation [1].

Electrostatic actuation is generated by applying electrical voltage between a microplate and a fixed electrode

plate being on the opposite side. In resonance microsensor applications, a combination of AC and DC

voltage is used. In these configurations, microplate deflects to static position due to DC electrostatic

actuation and then, AC actuation oscillates the system around that position. In this system, DC voltage

serves as the regulator of system sensitivity and natural frequency. The voltage in which a microplate

becomes unstable and connects to an opposite fixed electrode plate is called pull-in voltage [2]. Many

studies have been conducted for modeling and calculating the static deflection, pull-in voltage, natural

frequency and dynamic response of the system. The most important of them are addressed below.

Vogl and Nayfeh [3] established the equations governing the clamped circular plate, and then discretized

the system by using a Galerkin approach. They solved the equations for the equilibrium states due to a

general electric potential and determined the natural frequencies of the axisymmetric modes for the stable

deflected position. In another work, they investigated the response of the electrically actuated clamped

circular plate to a primary resonance excitation of its first axisymmetric mode using an analytical reduced-

order model [4]. Liao et al. indicated that the ratio of the dynamic to static pull-in voltages for a clamped

circular microplate is approximately 92%. Moreover, they demonstrated that, when the squeezed-film effect

induced by the air gap between the two plates is taken into account, the value of this ratio increases slightly

[5]. An electrostatically actuated microplate is modeled by Bertarelli et al. [6] for micropump application.

Wang et al. [7] analyzed the thermal and size effect on the nonlinear vibration of electrostatically actuated

circular microplate. It is shown in this work that geometrically nonlinear strain has a significant influence on

the frequency for the large initial gap to thickness ratio of the plate. The free vibration of symmetric circular

FML hybrid plates is studied by Shooshtari and Dalir [8]. They obtained the governing equation of motion

by considering von Karman nonlinearity and by using FSDT. Khorshidi et al. [9] analyzed static pull-in

instability and natural frequency of circular and annular plates in electrical field. They studied the effect of

rigid core, radial load, geometric nonlinearity, inner radius on pull-in instability of circular plates.

Karimzadeh et al. [10] studied the size dependent dynamic behavior of circular rings on elastic foundation

using modified couple stress theory. Mechanical behavior of the capacitive microphone with clamped

circular diaphragm is studied by Dowlati et al. [11] using modified couple stress theory. They obtained the

governing equation of motion by using Kirchhoff thin plate theory. Afterwards they used a SSLM method to

linears the equation. At, last they used Galerkin method to solve the equations.

Nonlinear vibration of rotating annular disc made of FGM with variable thickness is investigated by

Shahriari et al. [12]. They examined the natural frequencies and critical speed of the rotating FG annular

disk with two types of boundary conditions.

Jallouli et al. proved that how the von Kármán nonlinearity and the plate imperfections lead to a

significant delay in pull-in occurrence [13]. Saghir et al. [14] studied the static and dynamic behavior of an

imperfect plate under electrostatic actuation. They used von Karman in order to obtain governing equation.

Reduced order model based on Galerkin method is used to simulate mechanical behaviors of microplate.

They used experimental data in order to validate their work.

In some other studies, the pull-in instability and the vibration for the elliptic and circular electrostatically

actuated microplate are investigated in consideration of the Casimir force [15,16,17,18]. Zhang [19] used a

principle of virtual work to conduct the large deflection of a circular plate. He proposed a new approximate

analytical solution to study the plate-membrane transition behavior of the deflection. Medina et al.[20,21]

studied the behavior of initially curved circular microplates under electrostatic actuation. Their model was

based on Kirchhoff's hypothesis and the nonlinear von Kármán strain-displacement relations. Caruntu et al.

[22,23] studied the resonance of an electrostatically actuated circular plate consisting of deformable and

conductive circular plates in the presence of surface force. They showed that pull-in occurs at a relatively

large AC voltage. They used perturbation method to study the nonlinear parametric resonance. In another

study, the effects of mechanical shock on the stability and dynamic response of a MEMS circular capacitive

microphone was investigated [24]. It was shown that mechanical shock loads can induce considerable noise

in the response of the microphone. The mechanical behavior of a circular FGM micro-plate subjected to

electrostatic force and mechanical shock was studied by Sharafkhani et al.[25] They used a step-by-step

linearization and reduced order based on Galerkin method to solve the nonlinear equation of the static

deflection and dynamic motion. Shabani et al. investigated the dynamic behavior of a circular micro-plate

interacting with compressible fluid and excited by electrostatic force. They utilized Kirchhoff’s thin plate

theory for the actuating micro-plate and assumed inviscid for the operating fluid, and also derived the

eigenvalue problem of the coupled system using Fourier-Bessel expansion [26]. In some work, the

mechanical behavior of an electrostatically actuated circular microplate in the presence of hydrostatic force

was investigated. Li et al. studied the effects of both electrostatic force and uniform hydrostatic pressure on

the resonant frequency of a clamped circular microplate [27]. Nabian et al. [28] evaluated the instability of a

circular plate under non-uniform electrostatic pressure and uniform hydrostatic pressure. They linearized the

governing equations and solved them by using finite difference method. Jie[29] studied the behavior the

electrostatically actuated circular microplate in the presence of uniform hydrostatic pressure before and after

pull in. He used a two-fold method of bisection based on the shooting method to solve differential equations.

Moreover, He investigated the effect of different parameters on pull in and validated their result by the

results obtained in the previous studies. The dynamic pull-in instability and free vibration of circular

microplates subjected to combined hydrostatic and electrostatic forces were investigated taking into account

size effect through use of the strain gradient elasticity theory [30,31]. Rashvand et al. considered the effect

of the intrinsic length-scale on the stability and fundamental frequency of a fully clamped circular micro-

plate, which can be used as a RF MEMS resonator. A modified couple stress theory was utilized to model

the micro-plate, considering the variable length-scale parameter [32]. In [33] a closed-form solution was

presented for in-plane and out-of-plane free vibration of simply supported functionally graded (FG)

rectangular micro/nanoplates. Further, the dynamic response of a circular clamped micro-plate actuated by a

DC/AC non-linear electrostatic coupling force with considering the residual stress, hydrostatic pressure

acting on the upper surface, and squeeze-film damping generated by the air gap between the vibrating

micro-plate and fixed substrate was considered in [34].

Despite the studies mentioned above, the mechanical behavior of a partially two-layered circular

microplate under electrostatic actuation has not been examined so far, which is addressed in this study. This

configuration is important in two aspects. The natural frequency in a MEMS device is an indication of the

system performance speed. Therefore, it is investigated how natural frequency of an electrostatically

actuated microplate can be increased by compositing it partially. Also, pull-in and nonlinear shift of

resonance frequency are undesirable phenomena for the MEMS devices. Therefore, it is examined how these

phenomena can be optimized by compositing the microplate as partial. To this end, the nonlinear equations

of motion are derived through classical laminated plate theory (CLPT). In a previous study [35], it is shown

that using the CLPT method is an acceptable method that results in an almost exact outcome that matches

experimental data. Consequently, CLPT is used in this work to analyze the behavior of microplates.

Then, the static position and natural frequency of vibration around static position are obtained using

Galerkin approach. The linear mode shapes of non-uniform microplate i.e. a microplate coated as partial by

a second layer are used as comparison functions. The forced vibration equations around static position are

separated using Galerkin method, and solved by the multiple scale perturbation theory. The impact of the

changes in the second layer radius, assuming that its thickness remains constant, and the effect of changes in

the second layer thickness, assuming that its radius remains constant, are investigated on the mechanical

behavior of the system. At the end, the simultaneous change in the radius and thickness of the second layer,

with the assumption that its volume remains constant, are considered on the static and dynamic response of

the system.

2. Modeling and Formulation

The system consists of a circular microplate with the radius Ro under clamped boundary conditions. Another

layer with the radius Ri coats a part of the first layer. The plate is made of silicon. Electrostatic voltage of

ˆ ˆcos( )d aV V t is applied between the microplate and fixed electrode plate, which is placed at the distance d from

it. In this relation �� is time, Vd is DC voltage, Va and are amplitude and frequency of AC voltage, respectively.

Fig. 1 shows an overall view of the studied system.

The system consists of two sections considering Fig. 1. In section 1, microplate is two layered and in section 2

it is single layer. By applying Newton's second law on sheared element shown in (Fig. 2), the equation of motion

in the first section will be [25]:

(1)

ˆ ˆ 0ˆ ˆr rdN N N

dr r

21

1 1 2 2 2

ˆ( ) 0

ˆˆ ˆz z

dQ Q d wp h h

dr r dt

ˆ ˆ

ˆ ˆz

r rdM M MQ

dr r

Where 1w indicates the transverse deflection of microplate in two layered section and r is radius position of

element, Qz is shear force, rN and N are the component of net force in the radial and tangential directions,

rM

and M are the components of net moment in the radial and tangential directions. 1

is the density of first layer

and 2

is the mass density of the second layer, 1h is the thickness of the first layer and 2h is the thickness of

second layer and p indicates electrostatic force defined as follows[1]:

(2)

2

21

ˆ ˆ( cos( ))

ˆ2( )

d aV V tp

d w

Kirchhoff theory is used to calculate moment, normal force and shearing force of the cross section of system.

By considering middle area of the first layer as the reference layer and using CLPT theory for a balanced

isotropic plate the strain of in the cross section of microplate along the radius may be defined as the following

formula:

0 0ˆˆ ˆˆ ˆ

0 0( )

rr rr rz

(3)

Where, z is the distance from the reference layer. Moreover ˆˆrr and are the radial and tangential strains in

the point with distance z from reference layer, moreover 0ˆˆrr ( 0

) and 0r ( 0

) are the strain and curvature in

radial (tangential), respectively, which are calculated as follows:

Where 0u indicates the deflection along the radius. Showing the stress in the radial and tangential directions

for the kth layer by ˆˆkrr and k

, respectively, one can write:

0 0ˆˆ ˆˆ ˆˆ ˆ

0 0( ( ), 1,2

k krr rr rr r

k kk kQ Q z k

(5)

Where:

2

11,2

1(1 )

kkk

kk

EQ k

(6)

Where, kE is the elasticity module of the layer 1 and 2 and k is the Poisson’s ratio for them. Considering the

00ˆˆ

00

ˆ( )

ˆ

ˆ( )

ˆ

rr

u r

r

u r

r

(4)

21

02

ˆ

01

ˆˆ ( )

ˆ

ˆˆ1 ( )

ˆ ˆ

r

w r

r

w r

r r

definitions of strain and stress in equations 1 to 5, the net force N and moment M are defined as:

1

1

2 0 0ˆ ˆˆ ˆˆ ˆ

0 01

2 0 0ˆ ˆˆ ˆˆ ˆ

0 01

k

k

k

k

kzr rr rr r

kz

k

kzr rr rr r

kz

k

NN dz A B

N

MM zdz B F

M

(7)

Considering that 1 10 1,

2 2

h hz z and 2 2 1z h z . The coefficient matrix of A, B and F are defined as follows:

(8)

1

1

1

2

11 12

21 22 1

2

11 12

21 22 1

2

11 12 2

21 22 1

,

,

,

k

k

k

k

k

k

z

kz

k

z

kz

k

z

kz

k

A AA Q dz

A A

B BB Q z dz

B B

F FF Q z dz

F F

By combining equations (1), (3), (4) and (7) one obtains.

(9) 2 3 20 0 0 1 1 1

11 112 3 2 2

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ( ) ( ) ( )1 ( ) 1 ( ) 1 ( )0

ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ

u r u r u r w r w r w rA B

r r r r rr r r r

2 3 20 0 0 1 1 1

11 112 3 2 2

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ( ) ( ) ( )1 ( ) 1 ( ) 1 ( )

ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆz

u r u r u r w r w r w rB F Q

r r r r rr r r r

By combining equations (9), 0ˆ( )u r is omitted and the equation of deflection governing of system will be as

follows:

(10) 3 2 21 1 1 11 11 11

3 2 211

ˆ ˆ ˆˆ ˆ ˆ( ) 1 ( ) 1 ( )( )

ˆ ˆˆ ˆ ˆz

w r w r w r A F BQ

r r Ar r r

By substituting equation (10) into equation (1), and adding the effect of viscous damping, the equation of

motion governing two layered section of microplate will be as.

(11) 4 3 2 21 1 1 1 11 11 11

4 3 2 2 311

222 2

1 1 2 2 2 22

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ( , ) 2 ( , ) 1 ( , ) 1 ( , )( )

ˆ ˆˆ ˆ ˆ ˆ ˆ

ˆ ˆˆˆˆ ˆ ( cos( ))( , )ˆ ( )

ˆ ˆ ˆ2( )

d a

w r t w r t w r t w r t A F B

r r Ar r r r r

V V tw r t d wc h h

t dt d w

Where c is the coefficient of viscous damping. Setting h2=0 and replacing 1w by 2w in equation (1), equation

of motion in one-layered section of microplate will be as follows:

(12) 3 4 3 21 1 2 2 2 2

4 3 2 2 3

221 1

1 1 2 21

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ( , ) 2 ( , ) 1 ( , ) 1 ( , )

ˆ ˆˆ ˆ ˆ ˆ ˆ2

ˆ ˆˆˆˆ ˆ ( cos( ))( , )ˆ ( )

ˆ ˆ ˆ2( )

d a

E h w r t w r t w r t w r t

r rr r r r r

V V tw r t d wc h

t dt d w

Where, 2w is the deflection of microplate in one-layered section of microplate. Thus, equation governing the

system is defined as:

(13)

ˆ0 ir R

4 3 2 21 1 1 1 11 11 11

4 3 2 2 311

221 1

1 1 2 2 2 21

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ( , ) 2 ( , ) 1 ( , ) 1 ( , )( )

ˆ ˆˆ ˆ ˆ ˆ ˆ

ˆ ˆˆˆˆ ˆ ( cos( ))( , )ˆ ( )

ˆ ˆ ˆ2( )

d a

w r t w r t w r t w r t A F B

r r Ar r r r r

V V tw r t d wc h h

t dt d w

ˆi oR r R

3 4 3 21 1 2 2 2 2

4 3 2 2 3

222 2

1 1 2 22

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ( , ) 2 ( , ) 1 ( , ) 1 ( , )

ˆ ˆˆ ˆ ˆ ˆ ˆ2

ˆ ˆˆˆˆ ˆ ( cos( ))( , )ˆ ( )

ˆ ˆ ˆ2( )

d a

E h w r t w r t w r t w r t

r rr r r r r

V V tw r t d wc h

t dt d w

For analytical convenience, the following change in variable is applied to the equation of motion. ˆ

, 1,2ii

ww i

d

rr

R

tt

T

(14)

Where

2 0.510 ( )

hT R

D

31

212(1 )

EhD

(15)

In other word displacement term in the governing equation is normalized with respect to the initial gap height

between the plates and electrode plate, the radial position term r normalized concerning the plate radius, and

time is normalized with respect to the constant T. The efficiency of these variables is that maximum value for

displacement, i.e., contacting microplate to the electrode and also the maximum value coordinate system along

the radius would be equal to one. After these variables change a set of non-dimensional parameters are

obtained. Which are as below:

(16) 2 210

ˆ( )h

RD

40

2 5 0.51 1

ˆ12(1 )( )

Rc c

h D

31

212(1 )

EhD

211 11 11

11

A F BD

A

By substituting equations (14-16) into equation (13) the dimensionless form of motion equation will be as:

(17)

0 i

o

Rr

R

4 3 2 2

1 1 1 1 11 11 11

4 3 2 2 311

22 2 31 2 1

2 4 21 1

( , ) 2 ( , ) 1 ( , ) 1 ( , )( )

( cos( ))( , ) 2( )

2(1 )

d a

w r t w r t w r t w r t A F B

r r Ar r r r r

V V tw r t Dh d w D dc

t D h dt D R w

1i

o

Rr

R

4 3 22 2 2 2

4 3 2 2 3

22 32 2

2 4 22

( , ) 2 ( , ) 1 ( , ) 1 ( , )

( cos( ))( , ) 2

2(1 )

d a

w r t w r t w r t w r t

r rr r r r r

V V tw r t d w Ddc

t dt R w

In follow to make the equations simpler, we can use the term 4 ( , )iw r t which is defined as:

4 3 24

4 3 2 2 3

( , ) 2 ( , ) 1 ( , ) 1 ( , )( , ) i i i i

i

w r t w r t w r t w r tw r t

r rr r r r r

1,2i (18)

3. Static deflection

Static equation is solved by Galerkin method. To do this, comparison functions are required. These functions

are considered mode shapes of linear system, with taking into account the non-uniformity of cross section. To

obtain the mode shape of non-uniform microplate, deflection is considered as 1 1( , ) ( ) iiw tw r t r e and

2 2( , ) ( ) iiw tw r t r e , which 1 ( )i r is the function of mode shape at two layered section and 2 ( )i r is the function

of mode shape at the single layer section, Moreover iw is natural frequency of free vibration and i indicate the

mode number. By substituting these assumptions, into equation (17) and setting the external electrostatic force

and damping effect equal to zero, the differential equation governing the mode shape will be:

(19)

0 i

o

Rr

R

2 421

1

( ) 0i

Dhr

D h

1i

o

Rr

R

2 42 ( ) 0i r

Equation (19) is the Bessel's equation and its solution is as below:

2 22 1 2

1 1

2 23 4

1 1

( ) ( ) ( )

( ) ( ) 0

i m i m i

m i m i

Dh Dhr C J r C Y r

D h D h

Dh DhC I r C K r

D h D h

0 i

o

Rr

R

2 5 6 7 8( ) ( ) ( ) ( ) ( ) 0i m i m i m i m ir C J r C Y r C I r C K r 1i

o

Rr

R

(20)

Where, J is Bessel function of the first kind, Y is Bessel function of the second kind, I is refined Bessel

function of the first kind and K is refined Bessel function of the second kind.

To obtain the unknown coefficients, boundary conditions and continuity conditions at Ri are used considering

Fig. 3. These boundary conditions are as:

(21)

ˆ ˆ ˆ1 2 20ˆ ˆ ˆ( ) , ( ) 0, ( ) 0,

o or r R r Rw r finite w r w r

And the continuity conditions are:

(22) 1 2 1 2 1 2 1 2 1 2, , , ( ) ( ), ( ) ( )M M N N Q Q w Ri w Ri w Ri w Ri

Where as shown in Figure 3, 1 1 1( )( )M N Q and 2 2 2( )( )M N Q are moment(normal force)(shear force) in the cross

section which connects two layered and single layered parts, after applying equations (21) and (22) into equation

(20) and setting determinant of coefficient equal to zero, the natural frequency and mode shapes are obtained.

Now, putting time derivatives equal to zero in equation (17), the equation governing static deflection will be as:

(23)

0 i

o

Rr

R

2 34 2 2

1 1 4

2( )(1 ( ))s s d

D dw r w r V

D R

22 34

1 4 21

2( )

(1 ( ))

ds

s

VD dw r

D R w r

1i

o

Rr

R

34 2 2

2 2 4

2( )(1 ( ))s s d

Ddw r w r V

R

234

2 4 22

2( )

(1 ( ))

ds

s

VDdw r

R w r

Where, 1sw indicates the static deflection in two layered section and 2sw indicates the static deflection in single

layered section of microplate. The solution of (23) is assumed as 1 1

1

( )

m

s i i

i

w r a

and 2 2

1

( )

m

s i i

i

w r a

, where

1i and 2i have been obtained in former, m is the number of modes for solving the equation, and ia is the

unknown coefficients which are obtained by applying Galerkin method. Now the considered assumption is

substituted into equation (23), and then it is multiplied by 1 , 1..i j m after that the outcome is integrated from

r=0...1 which results to the below equations:

(24)

2 32 4 2

1 1 1 14

10 0

1 13

2 4 22 2 2 24

1

2( (1 ) ( ( )) ( ) ) ( )

2( (1 ) ( ( )) ( ) ) ( ) 0

i i

o o

i i

o o

R R

R Rm

i i i j d j

i

m

i i i i j d j

iR R

R R

D da a r r dr V r dr

D R

Dda a r r dr V r dr

R

4. Natural Frequency of vibration about static deflection

In equation (17), iw can be considered as the sum of dynamic deflection (wdi(r,t)) and static deflection (wsi(r)).

Thus, by putting 1 1 1ˆ ( , ) ( , ) ( )d sw r t w r t w r in first relation and 2 2 2

ˆ ( , ) ( , ) ( )d sw r t w r t w r in the second relation of

equation (17) and expanding electrostatic force around static position wsi(r), one obtains.

(25)

0 i

o

Rr

R

2

4 1 12 21 2

1 1

2 22 3

14 2 31 1

2 22 3

1 14 51 1

( , ) ( , )( , )

2 ( cos( )) 2( cos( ))2( ) ( , )

(1 ( )) (1 ( ))

3( cos( )) 4( cos( ))( , ) ( ,

(1 ( )) (1 ( ))

d dd

d a a d ad

s s

d a d ad d

s s

w r t w r tDh CDhw r t

D h D h tt

V V V t V V tD dw r t

D R w r w r

V V t V V tw r t w r

w r w r

)t

1i

o

Rr

R

24 2 2

2 2

2 22 3

24 2 32 2

2 22 3

2 24 52 2

( , ) ( , )( , )

2 ( cos( )) 2( cos( ))2( ) ( , )

(1 ( )) (1 ( ))

3( cos( )) 4( cos( ))( , ) ( , )

(1 ( )) (1 ( ))

d dd

d a a d ad

s s

d a d ad d

s s

w r t w r tw r t C

tt

V V V t V V tD dw r t

D R w r w r

V V t V V tw r t w r t

w r w r

Considering equation (25) the equation of linear free vibration of system about static deflection is as:

(26)

0 i

o

Rr

R

2 234 12

1 12 4 21 1

( , ) 4( , ) ( ) ( , )

(1 ( ))

d dd d

s

w r t VDh Ddw r t w r t

D h t R w r

1i

o

Rr

R

2 22 34 2

2 22 4 22

( , ) 4( , ) ( ) ( , )

(1 ( ))

d dd d

s

w r t VD dw r t w r t

t D R w r

To obtain the natural frequency of linear vibration of system about static deflection, n it is assumed that

1 1

1

( , ) ( ) ( )

m

d n n

n

w r t r q t

and 2 2

1

( , ) ( ) ( )

m

d n n

n

w r t r q t

. Which ( )nq t is time coordinate function correspond to free

vibration of the system. By substituting these assumption into (26), multiplying the outcome by 1m and 2m and

integrating on the whole of microplate radius , the differential equation governing natural frequency is obtained

as.

(27)

242

1 1 1 120 01 1 1

23

1 14 201 1

21 14

2 2 2 22

1 1

2 3

( )( ( ) ( ) ) ( ( ) ( )

4( ( ) ( ) )) ( )

(1 ( ))

( )( ( ) ( ) ) ( ( ) ( )

4

i i

o o

i

o

i i

o o

R RN N

R Rnn m n m

i i

R N

Rdn m n

s i

N N

nn m n mR R

i iR R

d q tDhr r dr r r dr

D h dt

VDdr r dr q t

R w r

d q tr r dr r r dr

dt

D d

2 1

2 24 22 1

0 1,..( ( ) ( ) )) ( )(1 ( )) i

o

N

dn m nR

s iR

Vr r dr q t

D R w rm N

Now the natural frequency may be obtained by applying a similar process which is used for obtaining the natural

frequency of a lumped mass system.

5. Dynamic solution at primary resonance

Dynamic response is considered as 1 11( , ) ( ) ( )dw r t p t r and 2 21( , ) ( ) ( )dw r t p t r , which p(t) is time coordinate

function. After substituting dynamic response into equation (25) and multiplying it by mode shape and integrating

the outcome on the whole of microplate radius the following equation results:

(28)

1 2 3 4 5

2 3 286 7 9

2 21011

3 2 21213 14

( ) ( ) ( ) cos( ) ( )

( ) ( ) ( ) ( )( cos( ))cos( )

( ) ( ) ( cos( ))cos( )

( ) ( ) ( cos( )) cos( )cos( )

a

a

a

a

a

a a

a

K p t K p t K p t K V t K p t

KK p t K p t K p t V t

V t

KK p t V t

V t

KK p t V t K V t

V t

Where, coefficients of Ki are defined as follows:

14 4

1 11 11 12 120

( ) ( ) ( ) ( )

i

o

i

o

R

R

R

R

K r r dr r r dr

12 22

2 11 120 1

( ) ( )

i

o

i

o

R

R

R

R

DhK r dr r dr

D h

12 22

3 11 120 1

( ) ( )

i

o

i

o

R

R

R

R

CDhK r dr C r dr

D h

(29)

2 2 1

4 11 124 2 20 1 1

2 2( ( )) ( )(1 ( )) (1 ( ))

i

o

i

o

R

R d d

Rs sR

V VD dK r dr r dr

D R w r w r

2 22 2 12 2

5 11 124 3 30 1 1

2 2( ( )) ( )(1 ( )) (1 ( ))

i

o

i

o

R

R d d

Rs sR

V VD dK r dr r dr

D R w r w r

2 22 2 13 3

6 11 124 4 40 1 1

3 3( ( )) ( )(1 ( )) (1 ( ))

i

o

i

o

R

R d d

Rs sR

V VD dK r dr r dr

D R w r w r

2 22 2 14 4

7 11 124 5 50 1 1

4 4( ( )) ( )(1 ( )) (1 ( ))

i

o

i

o

R

R d d

Rs sR

V VD dK r dr r dr

D R w r w r

2 22 2 12 2

8 11 124 3 30 1 1

4 4( ( )) ( )(1 ( )) (1 ( ))

i

o

i

o

R

R d d

Rs sR

V VD dK r dr r dr

D R w r w r

2 2 12 2

9 11 124 3 30 1 1

2 2( ( )) ( )(1 ( )) (1 ( ))

i

o

i

o

R

R

Rs sR

D dK r dr r dr

D R w r w r

2 2 13 3

10 11 124 4 40 1 1

6 6( ( )) ( )(1 ( )) (1 ( ))

i

o

i

o

R

R d d

Rs sR

V VD dK r dr r dr

D R w r w r

2 2 13 3

11 11 124 4 40 1 1

3 3( ( )) ( )(1 ( )) (1 ( ))

i

o

i

o

R

R

Rs sR

D dK r dr r dr

D R w r w r

2 2 14 4

12 11 124 5 50 1 1

8 8( ( )) ( )(1 ( )) (1 ( ))

i

o

i

o

R

R d d

Rs sR

V VD dK r dr r dr

D R w r w r

2 2 14 4

13 11 124 5 50 1 1

4 4( ( )) ( )(1 ( )) (1 ( ))

i

o

i

o

R

R

Rs sR

D dK r dr r dr

D R w r w r

2 2 1

14 11 124 2 20 1 1

1 1( ( )) ( )(1 ( )) (1 ( ))

i

o

i

o

R

R

Rs sR

D dK r dr r dr

D R w r w r

Assuming that nonlinear terms are of weaker order compared to linear terms, equation (28) can be solved with

the help of multiple scales method of perturbation theory, so it is assumed that:

(30) 2 31 0 1 2 2 0 1 3 0 1( ) ( , , ) ( , , ) ( , , )P t p T T T p T T T p T T T

Whereas, 0T t ,

1T t and 2

2T t are time scales and is a dimensionless bookkeeping parameter

representing the order of terms. Using chain derivatives, it can be written that:

(31)

20 1 2

dD D D

dt

22 2 2

0 0 1 0 2 122 (2 )

dD D D D D D

dt

Whereas, nD is the operator of ; 0,1,2n

dn

dT

To strike a balance between the terms of equation (28) the coefficient 3K resulted from viscous damping is

considered to be order 2 , moreover the AC actuation is considered to be order

3 . By substituting equation

(30) into equation (28) and separating terms with equal order of , the following relations are obtained:

Order 1( ):

(32) 21 5

1 0 1 2 1 0 1 2220

( , , ) ( , , ) 0K Kd

p T T T p T T TKdT

Order 2( ) :

(33) 2 21 5

2 0 1 2 2 0 1 2 1 0 1 222 00

261 0 1 2

2

( , , ) ( , , ) 2 ( , , )

( , , )

K Kd dp T T T p T T T p T T T

K dT TdT

Kp T T T

K

Order 3( ):

(34)

0 0

2 2 21 5

3 0 1 2 3 0 1 2 1 0 1 2 1 0 1 22 22 0 20 1

23

2 0 1 2 1 0 1 2

0 1 2 0

4

2 0

36 71 0 1 2 2 0 1 2 1 0 1 2

2 2

( , , ) ( , , ) 2 ( , , ) ( , , )

2 ( , , ) ( , , )

1( ( ))2

2( , , ) ( , , ) ( , , )

i T i Ta

K Kd d dp T T T p T T T p T T T p T T T

K dT TdT dT

Kd dp T T T p T T T

dT T K dT

K dV e e

K dT

K Kp T T T p T T T p T T T

K K

Assuming 1 5

2

K K

K

the solution of equation (32) will be as follows:

(35) 0 01 0 1 2 1 2 1 2( , , ) ( , ) ( , )

i T i Tp T T T A T T e A T T e

Whereas A (T1,T2) is a complex coefficient which is obtained by applying solvability condition, by substituting

equation (35) into equation (33):

(36) 0 0

222 6

2 0 1 2 2 0 1 2 1 2 1 221 20

61 2 1 2

2

( , , ) ( , , ) 2 ( , ) ( , )

2( , ) ( , )

i T i TKd dp T T T p T T T i e A T T A T T e

dT KdT

KA T T A T T CC

K

CC indicates conjugate of complex term. For solvability condition the coefficient of secular term in equation

(36), i.e. the coefficient of 0i Te

must become equal to zero:

(37) 0 0

1 2 1 2

1 1

( , ) ( , ) 0i T i Td d

A T T e A T T edT dT

It is concluded from equation (37) that A should only be a function of 2

T . Particular solution of equation (36)

is as:

(38) 0226 2 2

2 0 1 2 22 22

1 2 ( ) ( )( , , ) ( ( ) )

3

i TK A T A Tp T T T A T e CC

K

Now 1

p and 2p from equations (35) and (37) is substituted into equation (34), actuation frequency variation

is considered as 2 where is detuning parameter. In final the terms causing secular terms in equations

are retained, so one obtains:

(39) 0 0

0 0

22 3

3 0 1 2 3 0 1 2 2 222 20

26

74 2

22 2

( , , ) ( , , ) 2 ( ) ( )

10( )1

( ( )) ( 3 )2 3

i T i T

i T i Ta

Kd dp T T T p T T T i e A T i A T e

dT KdT

K

KK KV e e

K K

NST CC

NST represents all terms which are not secular. The term with coefficient of 0i Te

in equation (39) is secular

term. By putting it equal to zero and substituting 2( )A T as polar, 2( )2

1( )

2

i Ta T e

, and separating real part from

imaginary one, the result will be:

2 2 2

2

( ) ( ) sin( ( ))2

aFVda T a T T

dT

(40) 2 2 22 2 2 2

2 2

1( ) ( ) cos( ( )) ( )

8 ( )

aFVdT Sa T T a T

dT a T

Where

26

4 7 322 2 2 2

2 2 2

10( )

, ( ) ( ) , ( 3 ) ,2 3

a

K

K V K KKF T T T S

K K K

(41)

By putting 2

2

( )d

a TdT

and 2

2

( )d

TdT

equal to zero in equation (40), the equation governing equilibrium

solution amplitude of (a0) will be:

(42) 22 2 2 20

0 ( ) ( ) ( )2 2

a

sa Fa V

Equation (42) demonstrates that maximum amplitude of 0a occurs when the term inside second parentheses is

equal to zero, thus:

(43) 20sa

0

2Fa

Considering equation (43) the nonlinear resonance frequency will be as:

(44) 22

3 2

4sF

6. Results and discussion

In order to show the accuracy of the calculations, the thickness of the second layer is put equal to zero and the

static deflection is compared with the results obtained by Vogl and Nayfeh [3]. This comparison shown is in Fig.

4 indicates efficient conformity amongst the results.

In the following, the effect of the changes of the second layer radius, with the assumption that its thickness

remains constant, the effect of the change of the second layer, with the assumption that its radius remains

constant, and the simultaneous effect of changes in the radius and thickness, with the assumption that its volume

remains unchanged, are investigated on the mechanical behavior of the system (see Fig. 5). In this study, the

parameters are considered according to table 1.

Fig. 6 shows variations of the static deflection at the microplate center for different values of the second layer

radius, assuming that its thickness remains constant. It is demonstrated that when the second layer is deposited on

the entire area of the first layer, the system has the maximum static deflection and lowest pull-in voltage. The

pull-in voltage is the voltage in which the slop of the curve tends to infinity. It shows that by a decrease in the

value of the second layer radius, the static deflection increases while the pull-in voltage decreases. These changes

are due to the changes in flexural stiffness. It is obvious that an increase in the value in the second layer radius

causes an increase of the value of flexural stiffness, which in turn results in an increase in the value of static

deflection. Also since the electrostatic force in the right part of the equation (23) has 21( )s

w in its denominator,

i.e., by an increase of the value of static deflection, the value of electrostatic force increases as nonlinear.

Therefore, variation changes of the static deflection and pull-in voltage show a nonlinear behavior.

Since the electrostatic force is nonlinear, the pull-in voltages have increased 100% with fivefold increase of the

length of the second layer radius, reaching 34Volt from 17Volt. Considering Fig. 7, the value of natural frequency

goes up as the radius of the second layer increases. Considering linear terms of equation (28), these changes are

in direct relation with K1 and K5, where K1 is due to the flexural stiffness and K5 is caused by the electrostatic

force. Considering the equation (29), the term 31( )si

w appears in the denominator of the equation governing on

5K . Since the static deflection decreases as nonlinear with an increase of the value of the second layer radius,

therefore, the natural frequency also increases as nonlinear.

In Fig. 8, frequency response function is shown for different values of the second layer radius. It shows a

decrease in the value of amplitude peak and nonlinear shift of resonance with an increase of the value of the

second layer radius. This reduction is caused by decrease of the value of S and F and increase of the value of

natural frequency which are the terms of equation (42), (43) and (44). There are the terms of 21( )s

w and

51( )s

F w and 41( )s

w in the denominator of governing equation on 4

K , 7

K and 6

K . By an increase in the

value of the second layer radius, these terms are reduced due to decrease of the value of static deflection. So

considering equation (41) parameter S decreases which causes that the amplitude and nonlinear shift of resonance

frequency decreases.

Fig. 9 shows the variations of the static deflection as the thickness of the second layer changes assuming that

its radius remains unchanged. It denotes that static deflection decreases by an increase of the value of the second

layer thickness, and pull-in voltage reaches to 34V from 24V when the thickness increases 2.5 times. Considering

fig. 10, the system’s natural frequency, increases by increase of the value of second layer thickness, assuming that

the radius of the second layer remains constant. Fig. 11 shows steady state vibration amplitude against changes

between excitation frequency from natural frequency, for different values of the second layer radius. It’s

demonstrates that non-linear shift of resonance frequency and also amplitude at resonance frequency decreases by

an increase of the value of second layer thickness. These variations may be verified according to the discussion

presented in previous paragraphs for verifying of the mechanical behavior of system due to variation of the

second layer radius.

To reach the most optimal possible mode, variations of static deflection and natural frequency are studied with

simultaneous change in radius and thickness of the second layer assuming that its volume remains unchanged, see

fig. 5. Firstly it is assumed that the second layer is deposited on the whole of area of the first layer, then the radius

of the second layer is decreased and its thickness is increased. Figs. 12 and 13 demonstrate that static deflection,

natural frequency and the pull-in voltage are, respectively, reduced, increased, increased and then after a special

value of the second layer radius this behavior is reversed. These Figs. denotes that maximum value for the natural

frequency and minimum value for static deflection occur when the second layer covers 50% of the first layer.

Fig. 14 shows when the second layer covers 20 percent of the first layer, then by an increase of the value of the

second layer radius to 50 percent of the first layer radius, the amplitude and nonlinear shift of response frequency

decreases and by increase it to 80 percent these parameter are reversed once. In fact, as the radius of the second

layer decreases, flexural stiffness increases due to the increase in thickness of middle section of microplate which

has an important role in equivalent bending stiffness in system. According to the discussion of figs. 6,7 and 8

increasing the bending stiffness, decreases static deflection , increases natural frequency and pull-in voltage, and

decreases the nonlinear natural shift of resonance frequency. As the process goes on and the radius of second

layer decreases more, second layer becomes like a concentrated mass in the middle of microplate. In fact,

microplate becomes like a one-layered circular plate with a concentrated mass in its center. So the variation of the

mechanical behavior is reversed.

Conclusion

In the present study, the static deflection, natural frequency, pull-in voltage and frequency response function of

a two-layered clamped microplate subjected to electrostatic actuation have been studied. First, non-linear

equations of motion are derived by using classical laminated pate theory (CLPT)). Then, differential equations

governing the static deflection and frequency of free vibration around the system static position are solved with

help of Galerkin approach. Three mode shapes of a nonuniform clamped microplate are used as the comparison

function. Moreover, the equations of vibration around static position are separated using Galerkin method, and

then solved by the multiple scale perturbation theory.

In this study, the mechanical behavior of the system is examined for different values of the radius and

thickness of the second layer. It has been observed that an increase of the value of the second layer radius when

its thickness remains constant as well as an increase of the value of the second layer thickness when its radius

remains constant causes to increase the pull-in voltage and natural frequency, while results in the decreased static

deflection, non-linear shift of resonance frequency and steady static response amplitude.

It has been shown that when the second layer is deposited on the whole area of the first layer, by decreasing the

value and increasing the thickness of the first layer radius assuming that its volume remains constant, static

deflection decreases while natural frequency increases before an special value for the second layer radius; this

behavior is then reversed. The results show that the highest frequency and lowest static deflection occurs when

the second layer covers fifty percent of the first layer. This result can be used for designing high speed

microsensors.

REFERENCES

1. Younis, M I. MEMS “linear and nonlinear statics and dynamics” Springer, New York (2011).

2. Poloei, E., Zamanian, M. and Hosseini, S. A. A. “Nonlinear vibration analysis of an electrostatically excited micro

cantilever beam coated by viscoelastic layer with the aim of finding the modified configuration”. Structural

Engineering and Mechanics, 61(2), pp. 193-207(2017).

3. Vogl, G. W. and Nayfeh, A. H. “A reduced-order model for electrically actuated clamped circular plates. ”Journal of

Micromechanics and Microengineering, 15(4), 684(2005).

4. Vogl, G. W. and Nayfeh, A. H. “Primary resonance excitation of electrically actuated clamped circular

plates.” Nonlinear Dynamics, 47(1), pp.181-192(2007).

5. Liao, L. D., Chao, P. C., Huang, C. W., et al. “DC dynamic and static pull-in predictions and analysis for

electrostatically actuated clamped circular micro-plates based on a continuous model.” Journal of Micromechanics

and Microengineering, 20(2), pp.025013 (2009) .

6. Bertarelli, E., Colnago, A., Ardito, R., et al. “Modelling and characterization of circular microplate electrostatic

actuators for micropump applications. In Thermal, Mechanical and Multi-Physics Simulation and Experiments in

Microelectronics and Microsystems”, 16th International Conference on (pp. 1-7). IEEE(2015).

7. Wang, K. F., Wang, B., and Zhang, C. “Surface energy and thermal stress effect on nonlinear vibration of

electrostatically actuated circular micro-/nanoplates based on modified couple stress theory.” Acta

Mechanica, 228(1), 129-140(2017).

8. Shooshtari, A., and Dalir, M. A. “Nonlinear free vibration analysis of clamped circular fiber metal laminated

plates.” Scientia Iranica. Transaction B, Mechanical Engineering, 22(3), 813(2015).

9. Khorshidi Paji, M., Dardel, M., Pashaei, M. H., et al. “The effect of radial force on pull-in instability and frequency

of rigid core circular and annular plates subjected to electrostatic field.” Scientia Iranica, 25(4), pp.2111-

2129(2018).

10. Karimzadeh, A., Ahmadian, M. T., and Rahaeifard, M. “Effect of size dependency on in-plane vibration of circular

micro-rings.” Scientia Iranica, 24(4), pp.1996-2008(2017).

11. Dowlati, S., Rezazadeh, G., Afrang, S., et al. “An accurate study on capacitive microphone with circular diaphragm

using a higher order elasticity theory.” Latin American Journal of Solids and Structures, 13(4), pp.590-609(2016).

12. Shahriari, B., Zargar, O., Baghani, M, et al. “Free vibration analysis of rotating functionally graded annular disc of

variable thickness using generalized differential quadrature method.” Scientia Iranica, 25(2), pp.728-740(2018).

13. Jallouli, A., Kacem, N., Bourbon, G., et al. J. “Pull-in instability tuning in imperfect nonlinear circular microplates

under electrostatic actuation.” Physics Letters A, 380(46), pp.3886-3890(2016) .

14. Saghir, S., Bellaredj, M. L., Ramini, A., et al. Initially curved microplates under electrostatic actuation: theory and

experiment. Journal of Micromechanics and Microengineering, 26(9), pp.095004(2016).

15. Batra, R. C., Porfiri, M. and Spinello, D. “Vibrations and pull-in instabilities of microelectromechanical von

Kármán elliptic plates incorporating the Casimir force. ”Journal of Sound and Vibration, 315(4), pp.939-960(2008).

16. Batra, R. C., Porfiri, M., and Spinello, D. “Reduced-order models for microelectromechanical rectangular and

circular plates incorporating the Casimir force.” International Journal of Solids and Structures, 45(11), pp.3558-

3583(2008).

17. Wang, Y. G., Lin, W. H., Li, X. M, et al. “Bending and vibration of an electrostatically actuated circular microplate

in presence of Casimir force. ”Applied Mathematical Modelling, 35(5), pp.2348-2357(2011).

18. Yang, W. D., Kang, W. B., Wang, X. “Thermal and surface effects on the pull-in characteristics of circular nanoplate

NEMS actuator based on nonlocal elasticity theory”. Applied Mathematical Modelling, 43, pp.321-336(2017).

19. Zhang, Y. “Large deflection of clamped circular plate and accuracy of its approximate analytical solutions” Science

China Physics, Mechanics & Astronomy, 59(2), pp.624602(2016).

20. Medina, L., Gilat, R. and Krylov, S. “Bistable behavior of electrostatically actuated initially curved micro

plate”. Sensors and Actuators A: Physical, 248, pp.193-198(2016).

21. Medina, L., Gilat, R., and Krylov, S. Modeling strategies of electrostatically actuated initially curved bistable micro

plates. International Journal of Solids and Structures, pp.118, 1-13(2017).

22. Caruntu, D. I., and Oyervides, R. Frequency response reduced order model of primary resonance of electrostatically

actuated MEMS circular plate resonators. Communications in Nonlinear Science and Numerical Simulation, 43,

pp.261-270. (2017).

23. Caruntu, D. I., and Oyervides, R. Frequency response reduced order model of primary resonance of electrostatically

actuated MEMS circular plate resonators. Communications in Nonlinear Science and Numerical Simulation, 43,

pp.261-270(2017)..

24. Vahdat, A. S., Rezazadeh, G. and Afrang, S. “Improving response of a MEMS capacitive microphone filtering shock

noise. ”Microelectronics Journal, 42(5), pp.614-621(2011) .

25. Sharafkhani, N., Rezazadeh, G. and Shabani, R. “Study of mechanical behavior of circular FGM micro-plates under

nonlinear electrostatic and mechanical shock loadings. ”Acta Mechanica, 223(3), pp.579-591(2012) .

26. Shabani, R., Sharafkhani, N., Tariverdilo, S, et al. “Dynamic analysis of an electrostatically actuated circular micro-

plate interacting with compressible fluid.” Acta Mechanica, 224(9), pp.20-25(2013).

27. Li, Z., Zhao, L., Ye, Z., Wang, H., et al. “Resonant frequency analysis on an electrostatically actuated microplate

under uniform hydrostatic pressure.” Journal of Physics D: Applied Physics, 46(19), 195108(2013).

28. Nabian, A., Rezazadeh, G., Haddad-derafshi, M., et al. “Mechanical behavior of a circular micro plate subjected to

uniform hydrostatic and non-uniform electrostatic pressure.” Microsystem Technologies, 14(2), pp.235-240(2008).

29. Tian-Jie, C. “Model to Analyze Micro Circular Plate Subjected to Electrostatic Force” Sensors &

Transducers, 153(6), 129(2013).

30. Mohammadi, V., Ansari, R., Shojaei, M, et al. “Size-dependent dynamic pull-in instability of hydrostatically and

electrostatically actuated circular microplates.” Nonlinear Dynamics, 73(3), pp.1515-1526(2013).

31. Ansari, R., Gholami, R., Shojaei, M. F, et al. “Surface stress effect on the pull-in instability of circular

nanoplates. ”Acta Astronautica, 102, 140-150(2014).

32. Rashvand, K., Rezazadeh, G., Mobki, H., etal. “On the size-dependent behavior of a capacitive circular micro-plate

considering the variable length-scale parameter.” International Journal of Mechanical Sciences, 77, pp.333-

342(2013).

33. Salehipour, H., Nahvi, H., and Shahidi, A. R. “Exact analytical solution for free vibration of functionally graded

micro/nanoplates via three-dimensional nonlocal elasticity.” Physica E: Low-dimensional Systems and

Nanostructures, 66, pp.350-358(2015).

34. Liu, C. C. “The Stability and Non-linear Vibration Analysis of a Circular Clamped Microplate under Electrostatic

Actuation.” Smart Science, 5(3), pp.132-139(2017).

35. Deshpande, M. and Saggere, L. “An analytical model and working equations for static deflections of a circular

multi-layered diaphragm-type piezoelectric actuator.” Sensors and Actuators A: Physical, 136(2), pp.673-689(2007).

Captions:

Table 1. Parameters of system.

Fig. 1. Overall view of two layered microplate under AC-DC electrostatic actuation.

Fig. 2. Free body diagram for an element of microplate in two-layed section.

Fig. 3. Free body diagram at the boundary between single and two layered sections

Fig. 4. Static deflection variation of the center of microplate against changes in electrostatic voltage for one layer

microplate, h2=0.

Fig. 5. Increase of radius and decrease of thickness of the second layer by assuming that its volume remains constant.

Fig. 6. Variations of static deflection of the microplate center against the variation of electrostatic voltage for

different values of the second layer radius by assuming that second layer thickness remains constant (h2/h1=2/14).

Fig. 7. Variation of natural frequency of microplate against variation of electrostatic voltage for different values of the

second layer radius by assuming that second layer thickness remains constant (h2/h1=2/14).

Fig. 8. Variation of amplitude of steady state response against the variation of detuning parameter for different values

of the second layer radius assuming that its thickness remains constant (h2/h1=4/14, c=0.169).

Fig. 9. Changes in static deflection of the center of microplate against electrostatic voltage variations for different

values the second layer thickness by assuming that its radius remains constant (Ri/R0=0.5).

Fig. 10. Variation of the natural frequency of the system against electrostatic voltage variations for different values of

the second layer thickness by assuming that its radius remains constant (Ri/R0=0.5).

Fig. 11. Variations of amplitude of steady state response against the variation of detuning parameter for different values

of the second layer thickness assuming that its radius remains constant (Ri/R0=0.5, c=0.169).

Fig. 12. Variations of static deflection of the center of microplate against changes in electrostatic voltage for two layer

microplate with changes in radius and thickness by assuming that its volume remains constant.

Fig. 13. Variations of natural frequency against changes in electrostatic voltage for two layer microplate with changes

in radius and thickness by assuming the volume remains constant.

Fig. 14. Variations of vibration amplitude considering the amount of changes in σ with the variation of thickness and

radius by assuming the volume remains constant, , c=0.169.

Table 1. Parameters of system.

E h1 Vd Va

169 Gpa 140 μm 0.5 vol 0.03 vol

Fig. 1 Overall view of two layered microplate under AC-DC electrostatic actuation.

Fig. 2 Free body diagram for an element of microplate in two-layed section.

Fig. 3 Free body diagram at the boundary between single and two layered sections.

Fig. 4 Static deflection variation of the center of microplate against changes in electrostatic voltage for one

layer microplate, h2=0.

0 5 10 150

0.5

1

1.5

V2(volt)

Wm

ax

Present work

Previous work[3]

Fig. 5 Increase of radius and decrease of thickness of the second layer by assuming that its volume remains

constant.

Fig. 6 Variations of static deflection of the microplate center against the variation of electrostatic voltage for

different values of the second layer radius by assuming that second layer thickness remains constant

(h2/h1=2/14).

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

V2(volt)

Wm

ax

Ri/Ro=1

Ri/Ro=0.5

Ri/Ro=0.2

Fig. 7 Variation of natural frequency of microplate against variation of electrostatic voltage for different values of

the second layer radius by assuming that second layer thickness remains constant (h2/h1=2/14).

Fig. 8. Variation of amplitude of steady state response against the variation of detuning parameter for different

values of the second layer radius assuming that its thickness remains constant (h2/h1=4/14, c=0.169).

1 6 11 16 21 26 31 350

2

4

6

8

10

12

14

V2(volt)

Ri/Ro=1

Ri/Ro=0.5

Ri/Ro=0.2

Fig. 9 Changes in static deflection of the center of microplate against electrostatic voltage variations for different

values the second layer thickness by assuming that its radius remains constant (Ri/R0=0.5).

Fig. 10 Variation of the natural frequency of the system against electrostatic voltage variations for different values

of the second layer thickness by assuming that its radius remains constant (Ri/R0=0.5).

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

V2(volt)

Wm

ax

h2/h1=4/14

h2/h1=10/14

5 10 15 200

2

4

6

8

10

12

V2(volt)

h2/h1=2/14

h2/h1=10/14

Fig. 11 Variations of amplitude of steady state response against the variation of detuning parameter for different

values of the second layer thickness assuming that its radius remains constant (Ri/R0=0.5, c=0.169).

Fig. 12 Variations of static deflection of the center of microplate against changes in electrostatic voltage for two

layer microplate with changes in radius and thickness by assuming that its volume remains constant.

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

V2(volt)

W(r

)

Ri/Ro=1,h2/h1=1/14

Ri/Ro=0.2,h2/h1=25/14

Ri/Ro=0.5,h2/h1=4/14

Ri/Ro=0.8,h2/h1=15.625/140

Fig. 13 Variations of natural frequency against changes in electrostatic voltage for two layer microplate with

changes in radius and thickness by assuming the volume remains constant.

Fig. 14 Variations of vibration amplitude considering the amount of changes in σ with the variation of thickness and

radius by assuming the volume remains constant, , c=0.169.

5 10 15 20 25 300

2

4

6

8

10

12

14

V2(volt)

Ri/Ro=1,h2/h1=1/14

Ri/Ro=0.2,h2/h1=25/14

Ri/Ro=0.5,h2/h1=4/14

Ri/Ro=0.8,h2/h1=15.625/140

Reza Sepahvandi is an MSc student graduated from Mechanical Engineering Department, Kharazmi

University, Tehran, Iran. His research interests include: nonlinear vibration and MEMS/NEMS.

Mehdi Zamanian was born in Nahavand, Iran, in 1979. He received his B.S. degree from Razi

University, and M.S. and Ph.D. degrees in Mechanical Engineering from Tarbiat Modares

University,Tehran, Iran, and is currently associate professor in the Mechanical Engineering Department

of Kharazmi University, Tehran, Iran. His research interests include: nonlinear vibration, MEMS

devices and vortex induced vibration.

Behnam Firouzi is an MSc student graduated from Mechanical Engineering Department, Kharazmi

University, Tehran, Iran. His research interests are in the areas of nonlinear vibration, nonlinear Dynamics

and MEMS devices.

Seyyed Ali Asghar Hosseini was born in Qom, Iran, in 1978. He received his B.S. degree from Iran

University of Science and Technology, and M.S. and Ph.D. degrees from Tarbiat Modares University,

Tehran, Iran. He is currently associate professor in the Mechanical Engineering Department of Kharazmi

University, Tehran, Iran. His research interests include: nonlinear and random vibrations.