The e ect of radial force on pull-in instability and ...

Scientia Iranica B (2018) 25(4), 2111{2129

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringhttp://scientiairanica.sharif.edu

The e�ect of radial force on pull-in instability andfrequency of rigid core circular and annular platessubjected to electrostatic �eld

M. Khorshidi Paji, M. Dardel�, M.H. Pashaei, and R. Akbari Alashti

Department of Mechanical Engineering, Babol Noshirvani University of Technology, Shariati Street, Babol, P.O. Box 484, PostalCode: 47148-71167, Mazandaran, Iran.

Received 22 August 2016; received in revised form 20 February 2017; accepted 17 June 2017

KEYWORDSPull-in instability;Circular and annularplate;Rigid core;Radial load;Arc lengthcontinuation;Natural frequency.

Abstract. This study investigates static pull-in instability and frequency analysis ofcircular and annular plates in electrical �eld. The plate is modelled based on the classicalplate theory with nonlinear Von K�arm�an strain-displacement �eld. The governing equationof motion and boundary conditions were obtained using the Hamilton principle. Forthis purpose, potential and kinetic energies and the results achieved through radial andelectrostatic forces were obtained. Governing partial di�erential equations were reduced toordinary di�erential equations by Galerkin's method. Then, static pull-in instabilities ofclamped circular and annular plates with clamped-clamped and clamped-simply supportedboundary conditions were analyzed by the arc-length continuation method. The e�ects ofrigid core, radial load, geometric nonlinearity, inner radius, and boundary conditions onpull-in instability and frequency of the plate were studied.© 2018 Sharif University of Technology. All rights reserved.

1. Introduction

Micro-Electro-Mechanical Systems (MEMS) consist ofmicro-sized mechanical and electrical elements inte-grated into a common silicon substrate. Due to highfrequency, lightweight, small-sized, and low-energyconsumption, MEMS are used widely in many �elds.Di�erent types of actuation, such as thermal, magnetic,piezoelectric, and electrostatic actuation, are employedin MEMS. Nowadays, electrostatically actuated MEMSplay an important role in many industrial devicessuch as sensors, microphones, micro-actuators, andmicro-pumps. One of the most important problems

*. Corresponding author.E-mail addresses: [email protected] (M.Khorshidi Paji); [email protected] (M. Dardel);[email protected] (M.H. Pashaei); [email protected] (R.Akbari Alashti).

doi: 10.24200/sci.2017.4208

of these systems is nonlinear pull-in instability dueto the presence of electrostatic force, causing failureof devices. Thus, the study and prediction of suchinstability is necessary to the design of such systems.

Pull-in instability was observed experimentally byNathanson et al. [1] and Taylor [2]. This instabilityappears when the electrostatic force goes beyond theelastic restoring force of the structure; hence, thesubstrate is touched [3]. The critical value of voltagein which this instability occurs is represented as thepull-in voltage. If the rate of voltage variation isvery low and negligible, inertia has nearly no e�ect onthe microstructure's behavior; accordingly, the pull-involtage is called the static pull-in voltage. However,when the rate of voltage variation is considerable,the e�ect of inertia has to be included. The pull-in instability related to this situation is called thedynamic pull-in instability; the critical value of voltage,corresponding to the dynamic instability, is referred toas the dynamic pull-in voltage [4].

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2112 M. Khorshidi Paji et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 2111{2129

Static pull-in behaviors of di�erent structures ofMEMS is studied. In some studies, lumped models,such as one-Degree-Of-Freedom (DOF) mass springsystem model [5,6], are used. There are some studieson two DOFs lumped models [6-8]. In addition,there are some studies on pull-in instability of micro-beams [9-15]. Baghani et al. [16] studied the vibra-tion of electrostatically actuated double-clamped andsimply-supported micro-beams using Variational Iter-ation Method (VIM). Rahaeifard and Ahmadian [17]investigated the static de ection and pull-in instabilityof electrostatically actuated micro cantilevers basedon the strain gradient theory. Gholami et al. [18]studied the nonlinear pull-in instability of electro-statically actuated micro-switches based on Mindlin'sstrain gradient elasticity and the Timoshenko beamtheory. They solved the problem using the GeneralizedDi�erential Quadrature (GDQ) method and the pseudoarc-length continuation technique. Xiao et al. [19]developed a size-dependent model for electrostaticallyactuated micro-beam with piezoelectric layers attachedbased on a modi�ed couple stress theory.

However, there are limited studies on membraneand plate structures. There are some studies on pull-in instability of rectangular plates, some of which aredescribed here. Srinivas [20] employed the classicallinear plate theory to study the static pull-in instabilityof simply-supported rectangular plates. Mukherjee etal. [21] investigated the pull-in instability of cantileverplate using Finite-Element Method (FEM). Zhao etal. [22] presented a reduced-order model of electricallyactuated square micro-plate. Their model accountsfor the electric-force nonlinearity and the mid-planestretching of the plate. They found linear un-dampedvibration modes numerically, using the hierarchicalFEM, which are applicable to a Galerkin approxima-tion. Moghimi Zand et al. [23] developed a hybridFinite-Element Method (FEM) and Finite-Di�erenceMethod (FDM) to investigate contact phenomenon inmicro-plates actuated by ramp voltages. They utilizedthe hybrid FEM-FDM model to compute values ofcontact time and dynamic behavior of rectangularmulti-layer micro-plates. Wang et al. [24] investigatedthe e�ects of surface energy on the pull-in instabilityof electrostatically actuated rectangular micro/nano-plates based on the modi�ed couple stress theory.

Annular and circular plates have been receivingincreasing interest in MEMS community, especially inmicro-pump applications [25]. Some researchers havestudied pull-in instability of circular plates. Wanget al. [26] investigated the pull-in instability andvibration of a pre-stressed circular electrostaticallyactuated micro-plate, with consideration of the Casimirforce. They used von K�arm�an's nonlinear bendingtheory of thin plates. For static deformation of theplate, they obtained the pull-in parameters using the

shooting method and studied the small amplitude freevibration with respect to the pre-deformed bendingcon�guration. They examined the in uences of variousparameters, such as the initial gap, thickness ratio,Casimir e�ect, and pre-stress, on the pull-in insta-bility behavior and natural frequency. Soleymani etal. [27] studied the static pull-in instability of circularmicro-plate and the e�ect of residual stress on pull-in parameters using �nite-di�erence method. In theirmodel, they employed linear bending theory of thinplates and ignored the geometric nonlinearity. Nayfehet al. [28] presented reduced-order models to studypull-in instability of beams, rectangular, and circularplates.

In literature reviews, there are limited studieson pull-in instability of annular plates, the e�ect ofrigid core, and in-plane force. Hence, in this work,static pull-in behaviors of circular and annular platesmodelled are studied according to classical plate theoryusing nonlinear von K�arm�an strain-displacement rela-tion. This theory can account for large de ection ofplate subjected to electrostatic force. By using arc-length continuation method, the e�ects of di�erentstructural and geometrical parameters, radial force,and rigid core on pull-in instability are studied. Inaddition, the e�ects of electrostatic force, structuraland geometrical parameters on natural frequencies ofplate are determined.

In the following, at �rst, the formulation ofproblem consisting of equation of motion and boundaryconditions is presented. Then, the solution and arc-length continuation methods are described; results anddiscussion of the obtained equation are given. Finally,the last section concludes the paper.

2. Statement and formulation of the problem

The geometric structures of circular and annular platesare shown in Figure 1. In Figure 1(a), a circular plateof radius, R, and thickness, h, subjected to radialload, Pr, is depicted, and Figure 1(b) shows a circularplate with central rigid core. The circular plate hasradius, R, and inner radius of central core, b. Theplate is suspended above the in�nite ground plane withan initial gap, d. A positive potential di�erence, V ,between the two conductors causes the plate to de ect.

According to the classic plate theory, the displace-ment �eld of the plate can be stated in to the followingform [29]:

ur(r; �; z; t) = u(r; �; t)� z @w@r

;

u�(r; �; z; t) = v(r; �; t)� zr@w@�

;

uz(r; �; z; t) = w(r; �; t): (1)

M. Khorshidi Paji et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 2111{2129 2113

Figure 1. A schematic of electrostatically actuated plate: (a) Solid circular plate and (b) circular plate with central rigidcore.

Using Eq. (1) and von K�arm�an strain-displace-ment relation and neglecting in-plane displacements,the strain components are de�ned as follows [29]:

"rr =12

�@w@r

�2

� z @2w@r2 ;

"�� =1

2r2

�@w@�

�2

� zr

�@w@r

+1r@2w@�2

�;

2"r� =1r@w@r

@w@�� 2

zr

�@2w@r@�

� 1r@w@�

�;

"zz = "�z = "rz = 0: (2)

The stress and strain relations for isotropic plate aregiven as [29]:

�rr =E

1� �2 (�rr + �"��;

�� =E

1� �2 (�� + ��rr); �r� = 2G�r� ; (3)

where G = E2(1+�) .

The Hamilton principle is used to develop thegoverning equations of motion. For this purpose, thekinetic and potential energies of the system should beobtained. The kinetics and strain energies of T and �can be expressed respectively as follows [29]:

T =12

x0

h=2Z�h=2

�( _u2r+ _u2

�+ _u2z)dzrdrd�+Tcore; (4)

� =12

x0

h=2Z�h=2

(�rr�rr + �� + �r� r�)dzrdrd�;(5)

where 0 denotes the domain of the plate. In Eq. (4),Tcore is the kinetic energy of rigid core that can beexpressed as [30]:

Tcore =12Mcore

�@w@t

��r=b

�2

: (6)

Due to symmetry in geometry, boundary and loadingconditions of the considered plate structure in Figure 1,in the �rst mode of instability, the rigid core only hastransverse displacement and is not necessary to includethe e�ect of rotational kinetic energy of rigid core. Thework, Wp, of radial force, Pr, can be expressed as inthe following form [31]:

Wp =x0

Pr"rrrdrd�: (7)

The electrostatic force per unit area between twoconductive plates is given by [32]:

fes =12

�V 2

(d� w)2 ; (8)

where V is the applied DC voltage. The work ofelectrostatic force, Wfe, can be expressed as [33]:

Wfe =x0

feswrdrd�: (9)

Then, the total work is given as in the following form:

W = Wp +Wfe: (10)


It is assumed that the plate has structural damping.The work of distributed structural damping, Wc, canbe expressed as follows [33]:

Wc =x0

c@w@twrdrd�: (11)

By substituting Eqs. (4)-(11) into extended Hamiltonprinciple and setting the coe�cients of �w in the areaintegrand equal to zero, equation of motion will beobtained as follows (see the Appendix):

� @@r

�rNrr

@w@r

�� 1r@@�

�N��

@w@�

�� @@r

�Nr�

@w@�

�� @@�

�Nr�

@w@r

�� @2(rMrr)

@r2 +@M��

@r

� 1r@2M��

@�2 � 2@2Mr�

@r@�� 2r@Mr�

@�

� Prr�@2w@r2 +

1r@w@r

�+ rI0 �w + rc _w

I2@2

@t2

�@@r

�r@w@r

�+

1r@2w@�2

�� r�V 2

2(d�w)2 =0;(12)

where I0 = �h is the mass per unit area, and I2 = �h3

12is the rotary inertia. Boundary conditions are:�

rNrr@w@r

+Nr�@w@�

+@(rMrr)@r

�M��

+ rI2@3w@r@t2

+@Mr�

@�

�r2�w(r2; t) = 0;

�rNrr

@w@r

+Nr�@w@�

+@(rMrr)@r

�M��

+ rI2@3w@r@t2

+@Mr�

@�

�Mcore@2w(b)@t2

�r1�w(r1; t) = 0;

�N��2r

@w@�

+Nr�@w@r

+@@�

�M��

r

�+

2rMr�

+I2r@3w@t2@�

+@Mr�

@r

�@w

��2�1

= 0

�rMrr��@w@r

��r2r1

= 0;

�M��

r��w��

��2�1

= 0;

�2Mr��w�r

��2�1

��r2r1

+r2Zr1

@Mr�

@r

��2�1

�w��2�1dr

+�2Z�1

@Mr�

@�

��r2r1

�w��r2r1d� = 0; (13)

where r1 and r2 are inner and outer radii, and �1 and �2are the angles of circular sectors. For circular (annular)plate, r1 = 0(b), r2 = R, �1 = 0, and �2 = 2�. Further,Nrr, N��, and Nr� are in-plane forces and Mrr, M��,and Mr� are out-of-plane moments known as the stressresultants (see Figure 2). These terms are de�ned asfollows [29]:

Nrr =

h=2Z�h=2

�rrdz; N�� =

h=2Z�h=2

��dz;

Nr� =

h=2Z�h=2

�r�dz;

Mrr =

h=2Z�h=2

�rrzdz; M�� =

h=2Z�h=2

��zdz;

Mr� =

h=2Z�h=2

�r�zdz: (14)

By substituting Eqs. (2) and (3) into Eq. (14), these

Figure 2. In-plane and out-of-plane forces and momentson a plate element.


forces and moments will be obtained as follows:

Nrr =Eh

1� �2

"12

�@w@r

�2

+ �1

2r2

�@w@�

�2#;

N�� =Eh

1� �2

"1

2r2

�@w@�

�2

+ �12

�@w@r

�2#;

Nr� = Gh1r@w@r

@w@�

;

Mrr = � Eh3

12(1� �2)

"@2w@r2 +

�r

�@w@r

+1r@2w@�2

�#;

M�� =� Eh3

12(1� �2)

"1r

�@w@r

+1r@2w@�2

�+ �

@2w@r2

#;

Mr� = �Gh3 212r

�@2w@r@�

� 1r@w@�

�: (15)

Galerkin's method is used for discretizing partial dif-ferential equation and discretizing them to a systemof coupled nonlinear Ordinary Di�erential Equations(ODEs). In Galerkin's method, displacement is writ-ten in terms of time-dependent generalized coordinateand spatial (trial) function. The trial function mustsatisfy all geometric and natural boundary conditionsof the problem. For the considered problem, naturalboundary conditions are nonlinear; hence, selectingspatial function to satisfy these boundary conditionsis very hard. Hence, a better way to challengenonlinear natural boundary conditions is to includethe related work into the equation of motion. Withthis strategy in inducing the work done by naturalboundary conditions into equation of motion, it is onlynecessary for the trial function to satisfy geometricboundary conditions.

According to Eq. (12), the equation of motionhas a fractional term of r�V 2

2(d�w)2 . Hence, to remedythe complicated form of equation of motion, the wholeequation is multiplied by (d � w)2�wdr. Then, theobtained equation is integrated into the whole regionof plate, and as previously described, for satisfyingnatural boundary conditions, the work done by bound-ary terms is added to the weighted residual form ofequation of motion. The following motion equation,in terms of displacements, will be calculated. Thisequation can be calculated by considering an axi-symmetric solution:

r2Zr1

(� Eh

2(1� �2)

"�@w@r

�2 @w@r

+ 3r�@w@r

�2 @2w@r2

�+

Eh3

12(1� �2)@@r

�@2w@r2 + r

@3w@r3 + �

@2w@r2

�� Eh3

12(1� �2)

�� 1r2@w@r

+1r@2w@r2 + �

@3w@r3

�� Prr

�@2w@r2 +

1r@w@r

�+ rI0 �w + rc _w

� I2 @2

@t2

"@@r

�r@w@r

�#)(d� w)2�wdr

�r2Zr1

r�V 2

2�wdr+d2

"12Eh

1��2 r�@w@r

�2@w@r

�w

��r2r1

+Eh3

12(1� �2)r�@2w@r2 +

�r@w@r

��@w@r

��r2r1

+Eh3

12(1� �2)

"1r@w@r

+ �@2w@r2

#�w

��r2r1

� @@r

�r

Eh3

12(1� �2)

"@2w@r2 +

�r@w@r

��w��r2r1

#=0:(16)

The following dimensionless variables are introduced:

�r =rR; �w =

wd; �1 =

6d2

h2 ; �2 =�R4

2Dd3 ;

�V = �2V 2; �t =1R2

D�h

� 12

t;

Id =I2I0R2 ; �Pr =

R2

DPr; (17)

where D = Eh3

12(1��2) . The dimensionless nonlineargoverning equation of motion can be obtained in thefollowing form:

r2Zr1

(� �1

�@ �w@�r

�3

+ 3�r�@ �w@�r

�2 @2 �w@�r2

!

+

�r@4 �w@�r4 + 2

@3 �w@�r3 � 1

�r@2 �w@�r2 +

1�r2@ �w@�r

�� r �Pr

�@2 �w@�r2 +

1�r@ �w@�r

�+ �r

@2 �w@�t2

+ �c�r@ �w@�t


� Id @2

@�t2

�@ �w@�r

+ �r@2 �w@�r2

�)(1� �w)2� �wd�r

�Z r2

r1�r �V � �wd�r + �1�r

�@ �w@�r

�3

� �w

��r2r1

+ �r�@2 �w@�r2

+��r@ �w@�r

�@� �w@�r

��r2r1

+

"1�r@ �w@�r

+ �@2 �w@�r2

�� w��r2r1

� @@�r

��r�@2 �w@�r2 +

�r@ �w@�r

�� w

��r2r1

= 0: (18)

As mentioned previously, in order to discretize partialdi�erential motion equation to a system of coupledODEs, Galerkin's method is used. In Galerkin's pro-cedure, �w(�r; �t) can be written as follows:

�w(�r; �t) =XN

j=1qj(�t)�j(�r); (19)

where qj(�t) is the time-dependent generalized coordi-nates, �j(�r) is the jth axisymmetric linear un-dampedmode shape of the at circular and annular plate, andN represents the number of modes retained in thesolution. �j(�r) has the following general form [33].

�j(�r) =C1J0(�j�r) + C2Y0(�j�r)

+ C3I0(�j�r) + C4K0(�j�r): (20)

J0 and Y0 are Bessel functions of the �rst and secondskinds, respectively, and I0 and K0 are modi�ed Besselfunctions of the �rst and second kinds, respectively.C1; : : : ; C4 and � are determined from boundary con-ditions.

The following nonlinear system of ODEs can bederived by substituting Eq. (19) into Eq. (18):

NXj=1

qj

r2Zr1

�i

"�r�ivj +2�000j � 1

�r�00j +

1�r2�

0j� �Pr

��r�00j +�0j

�#d�r

+NXj=1

�qj

r2Zr1

�i�

�r�j � Id(�0j + �r�00j )�d�r

+ �c1Xj=1

_qj

r2Zr1

�r�i�jd�r � 2NXj=1

NXk=1

qjqk

r2Zr1

�i�

�r�ivj �k

+ 2�000j �k � 1�r�00j �k

+1�r2�

0j�k � �Pr(�r�00j �k + �0j�k)

�d�r

� 2NXj=1

NXk=1

qj �qk

r2Zr1

�i�

�r�j�k � Id(�j�0k

+ �r�j�00k)�d�r � 2�c

NXj=1

NXk=1

qj _qk

r2Zr1

�r�i�j�kd�r

+NXj=1

NXk=1

NXm=1

qjqkqm

r2Zr1

�i�

�r�ivj �k�m

+ 2�000j �k�m � 1�r�00j �k�m +

1�r2�

0j�k�m

� �1�0j�0k�0m � 3�1�r�00j �0k�0m � �Pr(�r�00j �k�m

+ �0j�k�m)�d�r +

NXj=1

NXk=1

NXm=1

qjqk�qm

r2Zr1

�i�

�r�j�k�m � Id(�j�k�0m + �r�j�k�00m)�d�r

+ �cNXj=1

NXk=1

NXm=1

qjqk _qm

r2Zr1

�r�i�j�k�md�r

+ 2�1

NXj=1

NXk=1

NXm=1

NXn=1

qjqkqmqn

r2Zr1

�i��0j�0k�0m�n

+ 3�r�00j �0k�0m�n�d�r

� �1

NXj=1

NXk=1

NXm=1

NXn=1

NXr=1

qjqkqmqnqr

r2Zr1

�i

��0j�0k�0m�n�r+3�r�00j �0k�0m�n�r

�d�r �

r2Zr1

�r �V �id�r

+ �1

NXj=1

NXk=1

NXm=1

qjqkqm�r�i�0j�0k�0m��r2r1

+NXj=1

qj�

�r�0i�000j + ��0i�0j +1�r�i�0j � �r�i�00j

� ��i�0j � �r�i�000j +��r�i�0j

��r2r1

= 0;(21)

with i = 1; :::; N ; in this equation, primes on � denotederivatives with respect to r.


Some necessary descriptions are required to solvethe equation of motion. Due to the application of elec-trostatic force in MEMS, time response analysis needsstringent convergence condition, and they are sti� fortime response analysis. Using arc length continuationmethod can obviate the need for su�ciently small-timesteps. Moreover, by multiplying both sides of equationinto (d� w)2, the sti�ness of the equation is reduced.

Eq. (21) is a set of nonlinear coupled equations.For solving nonlinear algebraic equations, most meth-ods are iterative, such as continuation methods. Thesemethods are used to compute approximate solutionsfor nonlinear systems with parameterized nonlinearequations. In [34], a practical guide for performing pa-rameter studies is introduced. Numerical continuationis a tool to study how the behavior of a dynamicalsystem changes as a function of parameters. Pathfollowing in combination with boundary value problemsolvers has an important role in the development ofdynamical systems [35]. Herein, the arc-length con-tinuation method is used. In arc-length continuationmethod, the continuation direction is tangent to thesolution branch.

To obtain de ection-voltage curves for nonlinearEq. (21), the arc-length continuation procedure is usedaccording to Figure 3 [36,37].

Let the de ection-voltage curve have the followingform:

F (u; �) = 0: (22)

In this method, the �rst step is to solve the systemof nonlinear equations of Eqs. (21) or (22) for �xedvalues of system parameters and a very small valueof applied voltage. This starting point is used asan initial guess to start the procedure of the arc-length continuation method, until reaching the turningpoint. At the turning point, the Jacobian determinantof Eq. (22) with respect to u, i.e. @F=@u, is zero,and it is necessary to reverse the direction of changein parameter. With this procedure, the de ection-voltage curve can be obtained with su�cient accuracy,i.e. good conformity between the results obtained bycontinuation method and direct solution of ordinaryequations by numerical methods such as Runge-Kutta.

Figure 3. The de ection-voltage curve [37].

Figure 4. Mode shapes of fully clamped circular plate.

In the following part, a brief discussion of requiredlinear mode shapes for Galerkin's method is presented.First, fully clamped circular plate is considered. Theboundary conditions for clamped circular plate are:

�(1) = 0;@�(1)@�r

= 0; �(0) <1: (23)

The Bessel function of the second kind becomes in�niteat r = 0; therefore, constants C2 and C4 must be zero.Thus, Eq. (20) is simpli�ed to:

�(�r) = C1J0(��r) + C3I0(��r): (24)

By substituting Eq. (24) into Eq. (23), the mode shapesof clamped circular plate can be obtained as follows:

�(�r) = J0(��r)� J0(�)J0(�)

I0(��r): (25)

By applying the second condition of Eqs. (23) to (25)and solving the obtained equation numerically, � canbe obtained. In this work, the MATLAB software isused to obtain � and linear undamped mode. Somemode shapes of clamped circular plate are plotted inFigure 4.

The boundary conditions associated with theannular plate clamped from inside and outside edgesare:

�(1) = 0;@�@�r

(1) = 0; �(�) = 0;@�@�r

(�) = 0;(26)

where � = ri=R is the dimensionless inner radius.For annular plate clamped from the inside edge andsimply supported from the outside edge, the boundaryconditions are:

�(1) = 0; Mr = D�@2�@�r2 + �

1r@�@�r

��r=1

= 0;

�(�) = 0;@�@�r

(�) = 0: (27)


Figure 5. Mode shapes of (a) clamped-clamped and (b)clamped plates with simply supported inner and outeredges on the annular plate; � = 0:1, � = 0:33.

By substituting Eq. (20) into Eqs. (26) and (27)and numerically solving the obtained equations, thelinear un-damped mode shapes of clamped-clampedand clamped-simple annular plates are determined.Figure 5 shows mode shapes of annular plates with thementioned boundary conditions.

For a circular plate with a central rigid core, theboundary conditions are:

�(1) = 0;@�@�r

(1) = 0;@�@�r

(�b) = 0;

2��@3�@�r3 +

1�r@2�@�r2 � 1

�r2@�@�r

��r=�b

=Mc

R�h�!2�(�b); (28)

where �b = b=R is the dimensionless radius of rigid core.Following the same mentioned procedure for obtaining

Figure 6. Mode shapes of clamped circular plate withcentral rigid core, �b = 0:2, � = 0:34.

the mode shapes of previous boundary conditions, themode shape of this case can be calculated; however,since it requires a lengthy calculation, it is ignored here.Figure 6 shows mode shapes of clamped circular platewith central rigid core for �b = 0:2.

3. Results and discussion

In this section, in order to compare the pull-in in-stability of plate for di�erent boundary conditions,Eq. (21) was solved for fully clamped circular plate,clamped-clamped, and clamped-simply supported an-nular plates. Since the inertia terms are not importantin static pull-in instability, for studding it, all time-dependent terms of Eq. (21) are ignored. At �rst,results of clamped circular plate were compared withthose presented in the literature; afterwards, staticpull-in behaviors of circular and annular plates withdi�erent inner radii and boundary conditions werestudied, and the e�ect of radial load on pull-in insta-bility of plates was investigated. Then, the e�ect ofapplied voltage on the fundamental natural frequencyof deformed plate at the state of equilibrium of theplate was studied. The e�ect of rigid core on the pull-in stability was also accurately studied.

3.1. Static pull-in behavior of circular andannular plates

In this section, the static pull-in behaviors of circularand annular plates were studied. To validate theproposed model, the pull-in results of clamped circularplate were compared with the results obtained in [38],and the obtained equilibrium curve is shown in Fig-ure 7. The geometric and material parameters of themodel were selected as Poisson's ratio (�) of 0.25, outerradius (R) of 50 �m, thickness (h) of 3 �m, initialgap of 0.894 �m, and permittivity of free space (�) of8:85� 10�12 F/m.


Figure 7. Equilibrium curve for clamped circular plate; comparison of (a) Ref. [38] and (b) the present work.

Figure 8. Equilibrium curves for (a) clamped-clamped, (b) simply supported from outside and clamped from insideannular plate with convergence of �1 = 24, � = 0:1.

The saddle-node bifurcation points in Figure 7show static pull-in instability in which stable andunstable branches of solution coincide with each other.The value of applied voltage corresponding to thesepoints is static pull-in voltage, and the slope of allcurves becomes in�nite at the pull-in point. As shownin Figure 7, the results converge as the number ofmodes increases; using six modes for discretizing inGalerkin's procedure can be su�cient in order to obtainaccurate results. As seen in Figure 7, using theproposed model with six modes, the estimated staticpull-in voltage is 14.2, while Vogl [38] reported 14.1as the static pull-in voltage of the plate; this showsthe validity of the obtained equation and the proposedsolution method.

To study the behaviors of annular plates in elec-trostatic �eld, at �rst, annular plate clamped from bothinner and outer radii is considered, and then staticbehaviors of annular plates clamped from inner andsimply supported from outer radii were studied. Forthis study, the geometric and material parameters ofannular plate were selected as Poisson's ratio (�) of

0.33, outer radius (R) of 50 �m, inner radius/outerradius ratio (�) of 0.1, 0.3, and 0.5, thickness (h) of1 �m, initial gap (d) of 2 �m, and permittivity of freespace (�) of 8:85� 10�12 F/m.

First, it is necessary to select an appropriatenumber of mode shapes to describe the transversedisplacement. In order to determine the number ofmodes to be retained in the Galerkin's procedureand to obtain accurate results, a convergence studywas performed. By solving Eq. (21) for clamped-clamped and clamped-simply supported annular plateswith � = 0:1 and �1 = 24, the pull-in behavior ofannular plates was obtained, as shown in Figure 8,which demonstrates that using six modes is su�cientto obtain accurate results of annular plate; therefore,the following results were obtained by using six modes.

Now, after validating the equation of motion andsolution method and obtaining the required numberof mode shapes necessary for the Galerkin's methodof solution, e�ects of di�erent parameters on pull-in instability were determined. Since more attentionis devoted to circular plate in the literature, herein,


Figure 9. Equilibrium curves for annular plate with �1 = 24: (a) Clamped-clamped and (b) clamped-simply supportedplates; e�ect of inner radius.

Figure 10. The e�ect of geometric nonlinearity on equilibrium curve for (a) clamped-clamped annular plate with � = 0:1and (b) fully clamped circular plate.

the main priority is given to annular plate. Resultsof clamped-clamped and clamped-simply supportedannular plates with di�erent inner radii are presentedin Figure 9.

The lower branch of the obtained equilibriumcurve is stable, while the upper branch is unstable.At pull-in point, a jump in the transverse de ectionof plate will occur, and system will settle to its upperstable equilibrium curve (which is not shown in this�gure). The distance between unstable and stablebranches of equilibrium curve shows the margin ofstability, since the greater the distance, the greaterthe margin of safety. As expected, the pull-in voltageof clamped plate is greater than that of clamped-simply supported plate. Figure 9 demonstrates thatby increasing the inner radius of annular plate, pull-involtage develops in bigger voltages and the static pull-in voltage increases due to an increase in sti�ness ofplate.

In order to study the e�ect of geometric non-linearity on the pull-in parameters, the dimensionless

Table 1. Static pull-in parameters of clamped-clampedannular plates with di�erent geometric nonlinearitycoe�cients.

� �1 �2V 2 �Wmax

0.1

12 119.1 0.394218 125.1 0.387624 129.9 0.379430 133.9 0.3711

0.3

12 366 0.454518 402.2 0.454424 433.5 0.448730 460.8 0.4413

de ection of plate versus applied voltage was plot-ted for di�erent geometric nonlinearity coe�cients inFigure 10. This �gure shows equilibrium curves forfully clamped circular and clamped-clamped annularplates with � = 0:1 and equilibrium curves for di�erentgeometric nonlinearities. Results of other cases arepresented in Table 1.


As seen in Table 1, by increasing �1, the staticpull-in voltage increases while the de ection in pull-involtage decreases.

3.2. E�ect of radial load on pull-in behaviorsof circular and annular plates

In this section, by applying tensile and compressiveradial forces to circular and annular plates, the e�ectsof these forces on pull-in parameters are studied. In- uences of the radial force on the static pull-in voltageof circular plate with speci�cations, as mentioned inFigure 7, are depicted in Figure 11 for clamped circularplate. Dimensionless applied loads ( �Pr) are written oneach curve. Negative numbers show compressive load,and positive ones show tensile load. As known fromelasticity and vibration studies, tensile load increasessti�ness, while compressive load decreases sti�ness;

Figure 11. The e�ect of radial load of equilibrium curvefor clamped circular plate.

accordingly, pull-in voltages increase for tensile loadand decreases for compressive load.

From this �gure, it can be understood that thevariation of pull-in voltage with radial load is verymuch, while maximum de ection at the pull-in pointhas small variation with radial load.

In Figure 12, equilibrium curves for clamped-clamped and clamped-simply supported annular plateswith � = 0:1 and �1 = 24 under di�erent radial loadsare shown. In this case, since annular plate has greatersti�ness with respect to circular plate, it has greaterpull-in voltages.

Results of annular plates with �1 = 24 anddi�erent values of � under various radial loads arepresented in Tables 2 and 3.

As shown in Figures 11 and 12 and Tables 2and 3, by increasing tensile radial force, the sti�ness ofplate increases; thus, the static pull-in voltage of plateincreases, while an increase in the value of compressiveradial force has softening e�ect on system and decreasesthe sti�ness, which, accordingly, decreases the staticpull-in voltage of plate.

3.3. Pull-in behavior of circular plate with thecentral rigid core

In this section, the static pull-in behavior of circularplate with the central rigid core is studied. Thegeometric and material parameters of circular plate andrigid core are mentioned in Table 4.

Similar to the previous cases, at �rst, the con-vergence of the solution of a clamped circular platewith the central rigid core is examined in terms of thedi�erent number of mode shapes. The obtained resultis shown in Figure 13. From this �gure, it is clear thatfour mode shapes are su�cient for the convergence ofsolution.

According to the results shown for circular andannular plate, since the form of equilibrium curve

Figure 12. The e�ect of radial load on equilibrium curve for annular plate with � = 0:1 and �1 = 24: (a)Clamped-clamped, and (b) clamped-simply supported plates.


Table 2. The e�ect of radial load on static pull-in voltages of clamped-simply annular plates with �1 = 24.

Radial loadDimensionless

radial load( �Pr)

Dimensionlesspull-in voltage

(�2V 2)

Max dimensionlessde ection(Wmax=d)

� = 0:1 � = 0:3 � = 0:1 � = 0:3

Compressive{30 68.58 347 0.3849 0.4608{20 90.31 376 0.3827 0.4569{10 110.7 404.8 0.381 0.45350 129.9 433.5 0.379 0.4487

Tensile10 148.2 462 0.3772 0.444720 165.6 490.4 0.3754 0.440730 182.4 518.7 0.3742 0.4367

Table 3. The e�ect of radial load on static pull-in voltages of clamped-simply supported annular plates with �1 = 24.

Radial loadDimensionless

radial load( �Pr)

Dimensionlesspull-in voltage

(�2V 2)

Max dimensionlessde ection(Wmax=d)

� = 0:1 � = 0:3 � = 0:1 � = 0:3

Compressive{15 73.45 408.1 0.4646 0.6889{10 83.99 426.1 0.4625 0.6881{5 94.41 444 0.4604 0.68730 104.7 462 0.4584 0.6865

Tensile

5 114.9 480 0.4564 0.685710 125 498 0.4544 0.684915 135 516.1 0.4524 0.684120 144.9 534.1 0.4505 0.6832

Table 4. The geometric and material parameters ofcircular plate with the central rigid core.

Parameters Values

Radius of plate (R) 50 �mRadius of core (b) 20 �mThickness of plate (h) 1 �mThickness of core (hc) 2 �mInitial gap (d) 2 �mDensity of plate (�) 2700 kg/m3

Density of core (�c) 7800 kg/m3

Poisson's ratio (�) 0.34

is similar in di�erent cases, herein, only one case isexamined. Figure 14 represents the results determinedfor clamped solid circular and clamped circular plateswith central rigid mass.

As shown in Figure 14, the attachment of rigidcore to di�erent radius ratios has di�erent e�ects onpull-in voltage. Generally, due to the attachmentof rigid core, there is a reduction in the sti�ness ofplate. Hence, due to the reduction in the sti�ness ofstructure, pull-in voltage reduces, which is the reasonfor the reduction in pull-in voltage in b=R = 0:1 and

Figure 13. Equilibrium curves for clamped circular platewith the central rigid core with the convergence of�1 = 24, b=R = 0:2.

b=R = 0:15. However, this trend is not monotone withan increase in the radius of rigid core. In circularplate without rigid core, the de ection at the centerof rigid core increases with applied voltage with no


Figure 14. Comparison of equilibrium curves for clampedcircular and circular plates with the central rigid core;�1 = 24.

limitation on it, since the whole structure has the samedeformation. However, when the rigid core is attachedto plate, there is limitation on the de ection at thecenter point, since all points in the domain of rigid corehave the same displacement. The limitation on thedisplacement of the rigid core will be more stringentwith an increase in the de ection of plate. Hence,although the plate with a rigid core has lower sti�nessat its �rst initial shape, its rigidity will continuouslyincrease with the de ection at center. This is the reasonfor an increase in pull-in voltage for b=R = 0:2.

3.4. Natural frequencies of plate in theelectrostatic �eld

When the plate is de ected, the natural frequencieschange correspondingly. With the increase in appliedvoltage, the sti�ness of plate reduces continuously,until the sti�ness in the �rst mode of pull-in instabil-ity becomes zero in pull-in voltage, and the naturalfrequency becomes zero, accordingly. Hence, in thefollowing, reduction in the natural frequency of thisdeformed plate at every de ection shape of plate dueto electrostatic force is determined. To study thelinear natural frequency of vibration, it is necessaryto linearize the plate equations of motion around thede ected position.

Figures 15-17 show the variation of fundamentalnatural frequency of the de ected plate, normalizedwith respect to the natural frequency at the unreformedstate along with the variation in the electrostatic loadfor di�erent ratios of �1 (i.e. the geometric nonlinearityterm); the radius ratio of � is considered, too.

As shown in Figures 15-17, for low values of �1,when electrostatic load increases, the fundamental nat-ural frequency decreases and approaches zero as pull-

Figure 15. Fundamental natural frequency of de ectedclamped circular plate versus applied voltage for variousvalues of �1.

in develops. For high values of �1, the fundamentalnatural frequency �rst increases with electrostatic loaddue to strain hardening, and then decreases and ap-proaches zero. In clamped-circular plate, the sti�nessmonotony decreases with an increase in applied voltageand, accordingly, natural frequency decreases. Forannular plate with smaller inner radius, the conditionis the same, while, for higher values of inner radius, thefundamental frequency of plate increases at �rst sincethere is limitation on the transverse displacement of theplate; �nally, at higher values of applied pull-in voltage,zero frequency occurs. Due to limitation on thetransverse displacement of annular plate with a biggerinner radius, when the value of �1 is increased, there isa pronounced hardening e�ect for higher values of �1and �, and natural frequency increases accordingly insome frequency ranges of applied voltage.

4. Conclusion

In this paper, the static pull-in behavior and naturalfrequency of vibration for electrostatically actuatedcircular and annular plates were investigated. It wasfound that increasing the tensile radial force increasesthe pull-in voltage, and increasing the compressiveradial force decreases it. Moreover, results showed thatincreasing the inner radius of annular plates causes anincrease in the static pull-in voltage; by increasing ge-ometric nonlinearity, pull-in voltage develops in biggervoltage. Subsequently, the e�ect of applied voltage onfundamental natural frequency of de ected plates wasinvestigated. Results showed that by increasing appliedvoltage, the fundamental natural frequency decreasesand approaches zero as pull-in voltage develops. The


Figure 16. Fundamental natural frequency of de ected clamped-clamped annular plate with various values of �1: (a)� = 0:1, and (b) � = 0:3.

Figure 17. Fundamental natural frequency of de ected clamped-simply supported annular plate for various �1: (a)� = 0:1, and (b) � = 0:3.

obtained results show that variations of di�erent pa-rameters have great in uence on pull-in voltage andsmall in uence on de ection at pulling point. Thismeans that de ection at the pull-in condition is nearlythe same for di�erent parameters of a plate and is morein uenced by the boundary condition with respect toother parameters of system.

NOMENCLATURER Radius of the plateh Thickness of the plated Gap between two electrodesb Radius of rigid corehc Thickness of rigid coreMcore Mass of rigid core" The dielectric constant of the gap

medium

t TimeE Young's modulusw Transverse displacement of neutral

axes in the z directionu; v In-plane displacementsI2 Rotary inertia�w Dimensionless transverse displacementD The exural rigidity of plate�1 Geometric structural nonlinearity

parameterId Related to the radius of gyrationNrr; N��;Nr� In-plane forcesT The kinetic energyTcore The kinetic energy of rigid core� Poisson's ratio


� Mass density of the plate�c Mass density of rigid coreG Shear modulus� Variational operatorPr Radial force per lengthfes Electrostatic force per unit areac Damping coe�cientV Applied voltageur; u�; uz Displacements in the r, �, and z

directionsI0 Mass per unit area�V Dimensionless voltage�r Dimensionless radius�t Dimensionless time�2 Electrostatic force parameter�Pr Dimensionless radial loadMrr;M��;Mr� Out of plane momentsII The strain energy0 Domain of the plate

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Appendix A

In this appendix, by employing extended Hamilton'sprinciple, Eq. (12) will be obtained. Extended Hamil-ton's principle is given as follows [29]:

t2Zt1

(�� W � �T )dt = 0: (A.1)

By substituting Eqs. (4)-(11) into Eq. (A.1), theextended Hamilton's principle can be written as:

t2Zt1

x0

h=2Z�h=2

(�rr��rr + �� + �r�� r�)dzrdrd�dt

�t2Zt1

x0

h=2Z�h=2

�( _ur� _ur + _u�� _u� + _uz� _uz)dzrdrd�dt

�t2Zt1

12Mcore

�@w@t

��r=b

�2

dt�t2Zt1

x0

fes�wrdrd�dt

�t2Zt1

x0

�(Pr�rr)rdrd�dt+�t2Zt1

x0

c@w@twrdrd�dt=0:

(A.2)

By using terms de�ned in Eq. (14), the variations of thekinetics and strain energy can be written as follows:


t2Zt1

��dt =t2Zt1

x0

h=2Z�h=2

(�rr��rr + �� + �r�� r�)

dzrdrd�dt =t2Zt1

x0

� h=2Z�h=2

�rr�@w@r

@�w@r

� z @2�w@r2

�dz +

h=2Z�h=2

��

1r2@w@�

@�w@�

� zr

�@�w@r

+1r@2�w@�2

��dz

+

h=2Z�h=2

�r��

1r@�w@r

@w@�

+1r@w@r

@�w@�

� 2zr

�@2�w@r@�

� 1r@�w@�

��dz�rdrd�dt

=t2Zt1

x0

(Nrr

�@w@r

@�w@r

��Mrr

@2�w@r2

+N��

1r2@w@�

@�w@�

�� M��

r

�@�w@r

+1r@2�w@�2

�+Nr�

�1r@�w@r

@w@�

+1r@w@r

@�w@�

��2

Mr�

r

�@2�w@r@�

� 1r@�w@�

�)rdrd�dt; (A.3)

t2Zt1

�Tdt =t2Zt1

x0

h=2Z�h=2

��

_ur� _ur + _u�� _u� + _uz� _uz�

dzrdrd�dt+t2Zt1

12Mcore

�@w@t

��r=b

�2

dt

=t2Zt1

x0

h=2Z�h=2

�� z @2w

@r@t

�� z @2�w

@r@t

�+�� zr@2w@t@�

�� zr@2�w@t@�

�

+ _w� _w�dzrdrd�dt+

t2Zt1

12Mcore

�@w@t

��r=b

�2

dt

=t2Zt1

x0

�I2@2w@r@t

@2�w@r@t

+I2r2@2w@t@�

@2�w@t@�

+ I0 _w� _w�rdrd�dt+

t2Zt1

12Mcore

�@w@t

��r=b

�2

dt:(A.4)

By substituting Eqs. (A.3) and (A.4) into Eq. (A.2),we obtain:

t2Zt1

x0

�Nrr

�@w@r

@�w@r

�+N��

�1r2@w@�

@�w@�

�+Nr�

�1r@�w@r

@w@�

+1r@w@r

@�w@�

��Mrr

@2�w@r2

� M��

r

�@�w@r

+1r@2�w@�2

�� 2

Mr�

r

�@2�w@r@�

� 1r@�w@�

��rdrd�dt

t2Zt1

x0

�I0 ( _w� _w)

+ I2�@2w@r@t

@2�w@r@t

+1r2@2w@t@�

@2�w@t@�

��rdrd�dt

�t2Zt1

x0

fes�wrdrd�dt�t2Zt1

x0

�(Pr�rr)rdrd�dt

+ �t2Zt1

x0

c@w@twrdrd�dt = 0:

(A.5)

By integrating Eq. (A.5) by parts in space and timeto relive �w from any di�erentiations, it is found thateach individual term simpli�es as follows:x

0

Nrr@w@r

@�w@r

rdrd�

=�2Z�1

rNrr@w@r

�w��r2r1d� �x

0

@@r

�rNrr

@w@r

��wdrd�;(A.6)

x0

N��1

2r2@w@�

@�w@�

rdrd�

=r2Zr1

N��2r

@w@�

�w��2�1dr �x

0

@@�

�N��2r

@w@�

��wdrd�;

(A.7)


x0

Nr�1r@w@�

@�w@r

rdrd�

=�2Z�1

Nr�@w@�

�w��r2r1d� �x

0

@@r

�Nr�

@w@�

��wdrd�;

(A.8)

x0

Nr�1r@w@r

@�w@�

rdrd�

=r2Zr1

Nr�@w@r

�w��2�1dr �x

0

@@�

�Nr�

@w@r

��wdrd�;

(A.9)

�x0

Mrr@2�w@r2 rdrd� = �

�2Z�1

rMrr��@w@r

��r2r1d�

+�2Z�1

@(rMrr)@r

�w��r2r1d� �x

0

@2(rMrr)@r2 �wdrd�;

(A.10)

�x0

M��

r@�w@r

rdrd� =��2Z�1

M��w��r2r1d�

+x0

@M��

@r�wdrd�; (A.11)

�x0

M��

r1r@2�w@�2 rdrd� =

�r2Zr1

M��

r��@w@�

��2�1dr +

r2Zr1

@@�

�M��

r

��w��2�1dr

�x0

@2

@�2

�M��

r

��wdrd�;

(A.12)

�2x0

Mr�1r

�@2�w@r@�

� 1r@�w@�

�rdrd� =

� 2r2Zr1

Mr��@w@r

��2�1dr

+ 2@Mr�

@��w��2�1

��r2r1

� 2x0

@2Mr�

@r@��wdrd�

+ 2r2Zr1

Mr�

r�w��2�1dr � 2

x0

1r@Mr�

@��wdrd�;

(A.13)

�2x0

Mr�1r@2�w@r@�

rdrd� =

�x0

Mr�@2�w@r@�

drd� �x0

Mr�@2�w@r@�

drd�

= �2Mr��@w@r

��2�1

��r2r1

+r2Zr1

@Mr�

@r

��2�1�w��2�1dr

+�2Z�1

@Mr�

@�

��r2r1�w��r2r1d� +

�2Z�1

@Mr�

@��w��r2r1d�

+r2Zr1

@Mr�

@r�w��2�1dr �x

0

2@2Mr�

@r@��wdrd�;

(A.14)

2x0

Mr�1r@�w@�

drd�

=r2Zr1

2rMr��w

��2�1dr�2

x0

1r@Mr�

@��wdrd�;

(A.15)

�t2Zt1

x0

I0 ( _w� _w) rdrd�dt =t2Zt1

x0

rI0 ( �w�w) drd�dt;(A.16)

�t2Zt1

x0

I2�@2w@r@t

@2�w@r@t

+1r2@2w@t@�

@2�w@t@�

�rdrd�dt

=Z t2

t1

� �2Z�1

rI2@3w@r@t2

�w��r2r1d�+

r2Zr1

1rI2

@3w@t2@�

�w��2�1dr

�x0

I2@2

@t2

�@@r

�r@w@r

�+

1r@2w@�2

��wdrd�dt

�:(A.17)

By substituting Eqs. (A.6)-(A.17) into Eq. (A.5)and setting each of the coe�cients of in the areaintegrand equal to zero, the �nal form of equation ofmotion and boundary conditions will be obtained asEqs. (12) and (13).

Biographies

Masoumeh Khorshidi Paji received BS and MSdegrees in Engineering Mechanics from Babol Noshir-vani University of Technology, Iran in 2012 and 2015,respectively. Her research interests include MEMSand NEMS, nonlinear vibrations and dynamics, smartstructures, and energy harvesting.


Morteza Dardel received a PhD degree from Amirk-abir University, Iran in Solid Mechanics in 2009, andis currently an Assistant Professor in the Facultyof Mechanical Engineering at Babol Noshirvani Uni-versity of Technology, Iran. His research interestsare aeroelasticity, nonlinear dynamics, vibration, andcontrol of continuous systems and smart structures.

Mohammad Hadi Pashaei received a PhD degreein Space Structures from the University of Surrey, UKin 2004, and is currently an Assistant Professor in theFaculty of Mechanical Engineering at Babol NoshirvaniUniversity of Technology, Iran. His research interestsinclude structural dynamics, damping in structures,and vibrational systems analysis.

Reza Akbari Alashti was born in Iran in 1963. Hereceived a PhD degree in Mechanical Engineering, Ap-plied Design, from Tarbiat Modares University, Tehran,Iran in 2006, focused on the limit load analysis ofcylindrical shells with opening under combined loadingusing constrained minimization techniques. Since 2006,he has been a faculty member in the Mechanical En-gineering Department of Babol Noshirvani Universityof Technology, Iran. Dr. Akbari Alashti is a memberof the Iranian Society of Mechanical Engineering,European Society of Mechanical Engineers, and theIranian Association of Naval Architecture and MarineEngineers. His research interests include stress analysisin shells and plates, nonlinear analysis of structures,limit load, and damage analysis.

The e ect of radial force on pull-in instability and ...

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