ORI GIN AL PA PER

Pull-in Voltage of Electrostatically-ActuatedMicrobeams in Terms of Lumped Model Pull-inVoltage Using Novel Design Corrective Coefficients

Ghader Rezazadeh • Mohammad Fathalilou •

Morteza Sadeghi

Received: 28 May 2009 / Revised: 10 September 2009 / Published online: 30 June 2011

� Springer Science+Business Media, LLC 2011

Abstract In this paper, we present a study of the static and dynamic responses of a

fixed–fixed and cantilever microbeam (using both the lumped and the distributed

models) to a DC and a step DC voltage. A Galerkin-based step by step linearization

method and a Galerkin-based reduced order model have been used to solve the

governing static and dynamic equations, respectively. The calculated static and

dynamic pull-in voltages have been validated by previous experimental and theo-

retical results and a good agreement has been achieved. The introduction of novel

design corrective coefficients, independent of the beam’s material and geometric

properties, results in a closed form relationship between static pull-in voltage of the

lumped model and static & dynamic pull-in voltages of the distributed models, and

takes into account the residual stresses, axial force and damping effects. Multiplying

these design coefficients with the static pull-in voltage of the lumped model, the

static and dynamic pull-in voltage of a given microbeam can be obtained without

the need to solve the nonlinear governing equations.

Keywords MEMS � Electrostatic actuation � Pull-in voltage � Design corrective

coefficients

1 Introduction

Due to recent advances in the technology of micro-electro-mechanical systems

(MEMS), micro-sensors and actuators driven by an electrostatic force have become

the focus of intensive study. These devices offer advantages due to their small size,

G. Rezazadeh (&)

Mechanical Engineering Department, Urmia University, Urmia, Iran

e-mail: g.rezazadeh@urmia.ac.ir

M. Fathalilou � M. Sadeghi

Mechanical Engineering Department, University of Tabriz, Tabriz, Iran

123

Sens Imaging (2011) 12:117–131

DOI 10.1007/s11220-011-0065-2

ease of production, low-energy consumption, and compatibility with the integrated

circuits (ICs). These micro-devices are the key components of many devices and

commonly seen in various applications such as pressure sensors [1, 2], micro-

mirrors [3], micro-pumps [4–6], accelerometer [7], and so forth. MEMS devices are

generally classified according to their actuation mechanisms. The most common

actuation mechanisms are electrostatic, pneumatic, thermal, and piezoelectric [8].

Electrostatically-actuated devices form a broad class of MEMS devices due to their

simplicity, as they require few mechanical components and low voltage levels for

actuation [8]. Microbeams (e.g., fixed–fixed and cantilever microbeams) are widely

used in many MEMS devices such as capacitive micro-switches and resonant micro-

sensors. These devices are to some extent, at a more mature stage than some other

MEMS devices. Fixed–fixed microbeams, due to their high natural frequencies, are

widely used in resonant sensors and actuators. One of the most important issues in

the electrostatically-actuated micro-devices is the pull-in instability. In many of the

microstructures, it is necessary to determine the pull-in voltage. The pull-in

phenomenon is divided broadly speaking into two branches: static and dynamic

pull-in instability. The static pull-in instability is a discontinuity related to the

interplay of the elastic and electrostatic forces. When a potential difference is

applied between a conducting structure and the ground level, the structure deforms

due to electrostatic forces. The elastic forces grow approximately with displace-

ment, whereas the electrostatic forces grow inversely proportional to the square of

the distance. When the voltage is increased, the displacement grows until, at some

point, the growth rate of the electrostatic force exceeds that of the elastic force and

the system cannot reach a force balance without a physical contact, thus pull-in

instability occurs. The critical voltage point is known as the static pull-in voltage.

Previous studies have predicted the pull-in phenomena based on static analysis by

considering the static application of a DC voltage [9–12]. In addition to the static

pull-in, some studies have also introduced a dynamic pull-in voltage [13, 14]. The

dynamic pull-in voltage is defined as a step DC voltage that, when applied suddenly,

leads to the instability of the system [13].

One of the useful practices in the scientific community is simplifying problems as

far as possible. For example, it is useful to develop solutions to problems in non-

dimensional form and to introduce dimensionless data that can be used as a reference

for similar classes of problems, independent of the geometric and material properties.

As mentioned earlier, there are a number of studies focused on the static and

dynamic pull-in phenomena, but there has not been introduced a relationship

between the static pull-in voltage of the lumped model of a microbeam to the static

and dynamic pull-in voltages of the distributed model by considering axial forces

and damping effects. The purpose of this paper is to introduce design corrective

coefficients for the static pull-in voltage of the lumped model of a cantilever and

fixed–fixed microbeams. These will allow the static and dynamic pull-in voltage of

the microbeams (considering axial forces and damping effects) to be calculated

without any need to solve the governing static and dynamic equations of the

distributed model. The coefficients can be used in all types of microbeams with

similar boundary conditions without any dependence on their geometric and

material properties. So, by presenting a mathematical model and numerical solution,

118 Sens Imaging (2011) 12:117–131

123

the response of cantilevers and fixed–fixed microbeams to static DC and step DC

voltage is calculated and validated using previous experimental and analytical

results. Then, novel dimensionless design corrective coefficients are introduced in

order to calculate the pull-in voltage of the microbeams.

2 Nonlinear Electromechanical Coupled Model

2.1 Model Description

A lumped model can be helpful for calculating the rough quantitative estimates of

the response of a wide range of electrostatically-actuated micro-structures. The

lumped model shown in Fig. 1 is utilized to represent a MEMS device employing

electrostatic actuation. The device has a movable microstructure of mass m, which

forms one side of a variable capacitor. A viscous damper of coefficient c is used to

model the energy dissipation and a spring of coefficient k is used to model the

effective stiffness of the microstructure due to the elastic restoring force [14].

The equivalent mass (m) and stiffness (k) of a lumped model for a microbeam are

calculated in the references. The equivalent mass (m) and stiffness (k) of lumped

model for a fixed–fixed microbeam are m = 0.41qbhL & k = 384EI/L3 and for a

cantilever microbeam are m = 0.26qbhL & k = 8EI/L3 respectively [15, 16].

Figure 2a, b shows an electrostatically-actuated fixed–fixed and cantilever

microbeam, respectively. The electrostatically-actuated microbeam is a suspended

elastic beam with an applied electrostatic force. The device consists of a beam,

suspended over a dielectric film deposited on top of the center conductor and fixed

at one or both ends (for cantilever and fixed–fixed microbeams, respectively) to the

Fig. 1 A lumped model of amicrobeam

Fig. 2 An electrostatically-actuated microbeam: a fixed–fixed, b cantilever

Sens Imaging (2011) 12:117–131 119

123

ground conductor. When a voltage is applied between the upper and lower

electrodes, the upper deformable beam is pulled down due to the electrical force.

The microbeams have width b, thickness h, length L, density q and with Young’s

modulus E. Parameters d, j and e are initial gap, dielectric coefficient of the gap

medium and dielectric constant, respectively. Let x denote the coordinate along the

length of the microbeams with its origin at the left end, and w the deflection of the

beams, defined to be positive downward (in the direction of z).

2.2 Mathematical Modeling

The equation of motion of the microbeam in the lumped model actuated by a step

DC voltage can be written as follows and the static equation is obtained by dropping

the time dependence from this equation [14].

m€zþ c _zþ kz ¼ eAV2ðtÞ2ðd � zÞ2

ð1Þ

where A is the electrode area on the microstructure.

In the distributed model, fixed–fixed microbeams represent an example of a

microstructure subject to a geometric nonlinearity generated by mid-plane

stretching. When a beam is in bending, the actual beam length L0 is longer than

the original length L, although there is no displacement in the x direction at the

beam ends. The actual length along the centerline of the beam is calculated by

integrating the arc length ds along the curved beam based on the cubic shape

functions for small deflection of the beam, w(x):

L0 ¼ZL

0

ds �ZL

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ dw

dx

� �2s

dx ð2Þ

Considering L � w, hence (dw/dx)2 � 1, and, as a result, the elongation is

approximately given by:

DL � 1

2

ZL

0

dw

dx

� �2

dx ð3Þ

therefore the stretching stress and force are given by:

ra ¼~E

2L

ZL

0

dw

dx

� �2

dx; and Na ¼ bhra ð4Þ

In the fixed–fixed microbeams, beside the stretching stress, residual stresses due

to the inconsistency of both the thermal expansion coefficient and the crystal lattice

period between the substrate and thin film, is unavoidable in surface micromachin-

ing techniques. Accurate and reliable data for residual stress are crucial to the proper

design of MEMS devices that are related to these techniques [17, 18]. Considering

the fabrication sequence of MEMS devices, the residual force can be expressed as:

120 Sens Imaging (2011) 12:117–131

123

Nr ¼ rr 1� tð Þbh ð5Þ

where rr is the biaxial effective residual stress [19].

Assuming the stretching and residual stresses effects, the governing differential

equation for static deflection of the fixed–fixed microbeam takes the following form

[13]:

~EId4w

dx4� Na þ Nr½ � d

2w

dx2¼ jeb

2

V

d � wðx; tÞ

� �2

ð6Þ

where I is the moment of inertia of the cross-sectional area and V is the applied

voltage to the parallel beams. For a wide beam which has b C 5h, the effective

modulus ~E can be approximated by the plate modulus E/(1 - m2); otherwise, ~E is

the Young’s modulus E [9]. The m is Poisson’s ratio of the beam material.

For convenience in analysis, Eq. 6 can take a nondimensional form, by

introducing nondimensional parameters as following:

w ¼ w

d; and x ¼ x

Lð7Þ

Substituting these parameters into Eq. 6, the following non-dimensional equation

is obtained:

d4w

dx4� Na þ Nr

� � d2w

dx2¼ V

1� wðx; tÞ

� �2

ð8Þ

where V is introduced as a dimensionless voltage and expressed as:

V ¼ffiffiffiffiffiffiffiffiffiffiffiffijebL4

2EId3

rV ð9Þ

The governing equation for the dynamic motion of the beam, w(x, t) actuated by

an electrical load of step DC voltage V(t) is written as [13].

~EIo4w

ox4þ qbh

o2w

ot2þ c

ow

ot� Na þ Nr½ � o

2w

ox2¼ jeb

2

VðtÞd � wðx; tÞ

� �2

ð10Þ

The microbeam can be subjected to a structural or a viscous damping [20, 21].

These effects are approximated by an equivalent damping coefficient, c per unit

length [20].

To nondimensionalize Eq. 10, beside the non-dimensional parameters in Eq. 6,

time is also nondimensionalized by introducing the characteristic period of the

beam, t* according to:

t ¼ t

t�; with t� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiqbhL4

~EI

rð11Þ

Substituting these parameters into Eq. 10, the following equation is obtained:

o4w

ox4þ o2w

ot2þ c

ow

ot� Na þ Nr

� � o2w

ox2¼ VðtÞ

1� wðx; tÞ

� �2

ð12Þ

Sens Imaging (2011) 12:117–131 121

123

where

c ¼ 12cL4

~Ebh3t�; Na ¼ 6

d

h

� �2Z1

0

ow

ox

� �2

dx; Nr ¼12NrL

2

~Ebh3ð13Þ

The fixed–fixed beam’s boundary conditions are given by:

w 0; tð Þ ¼ w 1; tð Þ ¼ 0 andow

ox0; tð Þ ¼ ow

ox1; tð Þ ¼ 0 ð14Þ

The static and dynamic governing equations of a cantilever microbeam are

obtained by dropping the stretching and residual stresses effects from Eqs. 8 and 12,

respectively, and the accompanying boundary conditions are:

w 0; tð Þ ¼ ow

ox0; tð Þ ¼ 0 and

o2w

ox21; tð Þ ¼ o3w

ox31; tð Þ ¼ 0 ð15Þ

3 Numerical Analysis

3.1 Distributed Model Equations

The numerical solution for the governing equation of the fixed–fixed and cantilever

beams is unique. Due to the nonlinearity of the derived static equation, the solution

is complicated and time consuming. Direct application of the Galerkin method or

finite difference method creates a set of nonlinear algebraic equations. In this paper

we used a method to solve it which consists of two steps. In first step, a step by step

linearization method (SSLM) [22] is used and in the second step, a Galerkin method

for solving the obtained linear equation is used.

In order to use SSLM, it is assumed that the wks , is the displacement of beam due

to the applied voltage Vk. Therefore, increasing the applied voltage to a new value,

the displacement can be written as:

wkþ1s ¼ wk

s þ dw ¼ wks þ wðxÞ ð16Þ

when

Vkþ1 ¼ Vk þ dV ð17ÞSo, the equation of the static deflection of the fixed–fixed microbeam (Eq. 8) can

be rewritten at step of k ? 1 as follows:

d4wkþ1s

dx4� Nkþ1

a þ Nr

� � d2wkþ1s

dx2¼ Vkþ1

1� wkþ1s ðxÞ

� �2

ð18Þ

Considering a small value of dV , it is expected that the w would be small enough,

hence using Calculus of Variations theory and Taylor’s series expansion about wk in

Eq. 17, and applying the truncation to first order for suitable values of dV , it is

possible to obtain the desired accuracy. The linearized equation to calculate w can

be expressed as:

122 Sens Imaging (2011) 12:117–131

123

d4wdx4� Nk

a þ dNa þ Nr

� � d2wdx2� dNa

d2w

dx2

����ðwk ;VkÞ

�2Vk� 2

1� wks

� 3w� 2

VkdV

1� wks

� 2¼ 0

ð19Þ

where variation of the hardening term based on Calculus of Variations Theory can

be expressed as :

dNa ¼Z1

0

wðxÞd2w

dx2

����ðwk ;VkÞ

dx ð20Þ

Considering a small value of dV and as a result w xð Þ, multiplying dNa with

d2w=dx2 would be small enough and can be neglected.

The obtained linear differential equation is solved by the Galerkin method. w xð Þbased on function spaces can be expressed as:

wðxÞ ¼X1j¼1

aj/j xð Þ ð21Þ

In this paper, /j xð Þ is selected as jth undamped linear mode shape of the straight

microbeam. The unknown w xð Þ, is approximated by truncating the summation series

to a finite number, n:

wnðxÞ ¼Xn

j¼1

aj/j xð Þ ð22Þ

Substituting Eq. 21 into Eq. 18, and multiplying by /i xð Þ as a weight function in

the Galerkin method and then integrating the outcome from x ¼ 0 to 1, a set of

linear algebraic equation is generated as:

Xn

j¼1

Kijaj ¼ Fi i ¼ 1; . . .; n ð23Þ

where Kij ¼ Kmij þ Ka

ij � Keij and:

Kmij ¼

Z1

0

/i/ivj dx Ka

ij ¼�Z1

0

/i ðNka þ NrÞ/00j þ

Z1

0

/j

d2w

dx2

����Wk ;Vk

dx

24

35d2w

dx2

����Wk ;Vk

0@

1Adx

Keij ¼2

Vk� 2

1� wks

� 3

Z1

0

/i/jdx Fi ¼ 2Vk

1� wks

� 2Vkþ1� Vk� Z1

0

/i xð Þdx

ð24Þ

To study the fixed–fixed microbeam response to dynamic loading a Galerkin-based

reduced order model can be used [23]. Due to the non-linearity of the electrostatic

force and the stretching terms, direct application of the reduced order model to

dynamic equation (Eq. 12) leads to the generation of n nonlinear coupled ordinary

Sens Imaging (2011) 12:117–131 123

123

differential equations and consequently the solution is more complicated. To

overcome this difficulty, the hardening (Na) and forcing terms in Eq. 12 are con-

sidered constant terms in each time step of integration and take the value of the

previous step. By selecting sufficiently small time steps, this assumption leads to

sufficiently accurate results. Eq. 12 can then be rewritten as following:

o4w

ox4þ o2w

ot2þ c

ow

ot� N

$

a þ Nr

�o2w

ox2¼ FðV ; w

$Þ ð25Þ

To achieve a reduced order model, wðx; tÞ can be approximated as:

w x; tð Þ ¼Xn

j¼1

Tj tð Þ/j xð Þ ð26Þ

By substituting Eq. 26 into Eq. 25 and multiplying by /i xð Þ as a weight function in

the Galerkin method and then integrating the outcome from x ¼ 0 to 1, the Galerkin

based reduced order model is generated as:

Xn

j¼1

Mij€TjðtÞ þ

Xn

j¼1

Cij_TjðtÞ þ

Xn

j¼1

ðKmij þ Ka

ijÞTjðtÞ ¼ Fi ð27Þ

where M, C, Km and Ka are mass, damping, mechanical and axial stiffness matrices,

respectively. Also F introduces the forcing vector. These matrices and vector are

given by:

Mij ¼Z1

0

/i/jdx Cij ¼ c

Z1

0

/i/jdx i; j ¼ 1; . . .; n

Kmij ¼

ð28Þ

Now, Eq. 27 can be integrated over time by various integration methods such as

the Runge–Kutta method where N$

a and w$ðx; tÞ in each time step of integration take

the value of the previous step.

The proposed procedures for a fixed–fixed microbeam are used to solve the static

and dynamic equations of a cantilever microbeam where there are no stretching and

residual stresses.

3.2 Design Corrective Coefficients

Using the numerical solution presented in the previous section, the static and

dynamic pull-in voltage of the microbeams can be calculated. In this section our

objective is to derive a relationship between static pull-in voltage of the lumped

model and the static & dynamic pull-in voltage of the distributed model for a fixed–

fixed and a cantilever microbeam considering axial forces and damping effects.

First, the static pull in-voltage of the lumped model for a microbeam is given as

follows [24]:

124 Sens Imaging (2011) 12:117–131

123

VLpull�in ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi8kd3

27jeA

rð29Þ

Using stiffness, k, for the fixed–fixed and cantilever microbeam, the pull-in

voltage can be obtained as follows:

For a fixed� fixed microbeam : VLpull�in ¼ 3:08

ffiffiffiffiffiffiffiffiffiffiffiffiEh3d3

jeL4

r

For a cantilever microbeam : VLpull�in ¼ 0:44

ffiffiffiffiffiffiffiffiffiffiffiffiEh3d3

jeL4

r ð30Þ

The relationship of the pull-in voltage of the distributed models to the lumped

model can be expressed by including several design corrective coefficients as

follows:

Vspull�in ¼ Vs

pull�in

ffiffiffiffiffiffiffiffiffiffiffiffiEh3d3

6jeL4

r¼ asbscs VL

pull�in

�

Vdpull�in ¼ Vd

pull�in

ffiffiffiffiffiffiffiffiffiffiffiffiEh3d3

6jeL4

r¼ adbdcdk VL

pull�in

� ð31Þ

where a indicates the dimensionless model correction coefficient. Coefficients b, cand k introduce the dimensionless stretching stress, residual stress and damping

design corrective coefficients, respectively. The subscripts s and d indicate static

and dynamic problems, respectively.

The design corrective coefficients in general can be obtained using existing

experimental results or results from a numerical analysis for a distributed model and

applying a least squares regression or a curve fitting method.

4 Results and Discussion

4.1 Model Validation

For verification of our numerical solution, a microbeam with the geometric and

material properties listed in Table 1 was considered.

In Tables 2 and 3 the calculated pull-in voltage was compared to the results of

existing work for the fixed–fixed and cantilever microbeams having properties

shown in Table 1. As shown the calculated pull-in voltages are in good agreement

with the results presented in previous work.

For validation of dynamic results with the results presented in the previous

works, a fixed–fixed microbeam was considered with the specifications of a pressure

sensor used by Hung and Senturia [26]: E = 149 GPa, q = 2330 kg/m3,

L = 610 lm, b = 40 lm, h = 2.2 lm and d = 2.3 lm. Because h is given as a

nominal value, it was modified to match the experimental pull-in voltage.

Accordingly, we set h = 2.135 lm. We assumed a residual stress of -3.7 MPa.

In Fig. 3, the calculated pull-in time obtained using the proposed method is

compared with the theoretical and experimental results of Hung and Senturia [26]

Sens Imaging (2011) 12:117–131 125

123

Table 1 The values of design

variablesDesign variable Value

B 50 lm

H 3 lm

D 1 lm

E 169 GPa

q 2331 kg/m3

e 8.85 PF/m

m 0.06

Table 2 Comparison of the pull-in voltage for a fixed–fixed microbeam

Residual

stress (MPa)

Our results

(V)

Energy model

(V) [11]

MEMCAD

(V) [11]

L = 350 lm 0 20.1 20.2 20.3

100 35.3 35.4 35.8

-25 13.8 13.8 13.7

L = 250 lm 0 39.5 39.5 40.1

100 57.3 56.9 57.6

-25 33.4 33.7 33.6

Table 3 Comparison of the pull-in voltage for a cantilever microbeam (L = 150 lm)

Our result CoSolve simulation [25] Closed form 2D model [25]

Pull-in voltage (V) 17.0 16.9 16.8

Fig. 3 Comparison of the pull-in time for no damping case without the stretching effects

126 Sens Imaging (2011) 12:117–131

123

for various values of step DC voltage. The pull-in time was found by monitoring the

beam response over time for a sudden rise in the displacement; at that point, the

corresponding time was reported as the pull-in time [13].

As Fig. 3 illustrates, calculated results are in excellent agreement with the

theoretical and experimental results. It was shown that, for the no damping case

when V is less than 8.18 V, the pull-in instability does not occur, so this step DC

voltage can be taken as the dynamic pull-in voltage for the microbeam.

4.2 Design Corrective Coefficients

Using Eqs. 9, 30, 31 and the presented numerical analysis, Tables 4 and 5 list the

static and dynamic pull-in design corrective coefficients, independent of the

geometric and material properties of the microbeams. The parameter f in Table 5 is

the nondimensional damping ratio. It is shown that the model correction factor, a,

for a fixed–fixed microbeam is less than for a cantilever, so it can be concluded that

the pull-in voltage in a lumped model for a fixed–fixed microbeam gives more

accurate results than for a cantilever. Also the dynamic model correction factor is

about 91% of the static model correction factor. This is in good agreement with

previous reports [27, 28] which indicated that the dynamic pull-in voltage can be as

low as 92% of the static pull-in voltage.

As shown in the nondimensional form of governing equations, the static and

dynamic pull-in voltages for fixed–fixed microbeams vary by the value of stretching

and residual stresses, so their design corrective coefficient b and c, are approximated

by a polynomial expression, using a Least Squares method (LEM) and plotted in

Figs. 4 and 5. Also, in the dynamic part we developed a polynomial expression for

the damping design corrective coefficient and plotted in Fig. 6. Testing of the results

shows that these polynomial expressions can predict the pull-in voltage accurately.

As shown in Fig. 4, the stretching has more effect on the dynamic pull-in than the

static pull-in especially at high values of d/h.

Table 4 Static pull-in design corrective coefficients for fixed–fixed and cantilever microbeams

Fixed–fixed microbeam Cantilever microbeam

as 1.1116 1.2087

bs 0.0247(d/h)2 - 0.0071(d/h) ? 1.0019 1

cs �0:0001N2r þ 0:0121Nr þ 1 1

Table 5 Dynamic pull-in design corrective coefficients for fixed–fixed and cantilever microbeams

Fixed–fixed Cantilever

ad 1.0124 1.0940

bd 0.0745(d/h)2 - 0.0105(d/h) ? 1.0013 1

cd �0:0001N2r þ 0:0122Nr þ 1 1

k �0:2386f2 þ 0:4119fþ 1 �1:1444f2 þ 0:5314fþ 1

Sens Imaging (2011) 12:117–131 127

123

4.3 Formulas Verification

For verification of the presented formulas, we considered a fixed–fixed and

cantilever microbeam used by Nayfeh et al. [29] and Younis et al. [14], respectively

where they have calculated the static pull-in voltage numerically and indicated that

the dynamic pull-in voltage for the no damping case is as low as 92% of the static

pull-in voltage. The microbeams specifications are:

Fig. 4 The variation of the stretching design corrective coefficient versus d/h in the fixed–fixedmicrobeams

Fig. 5 The variation of the residual stress design corrective coefficient versus nondimensional residualforce in the fixed–fixed microbeams

128 Sens Imaging (2011) 12:117–131

123

L = 510 lm, b = 100 lm, h = 1.5 lm, d = 1.18 lm, N ¼ 8:7 and

L = 100 lm, b = 10 lm, h = 0.1 lm, d = 2 lm, respectively.

The results are compared and given in Table 6. As shown there is an excellent

agreement between the results.

Therefore, in the pull-in analysis of the cantilever and fixed–fixed microbeams, it

is sufficient to multiply the design corrective coefficients obtained from Tables 4

and 5 and Figs. 4, 5 and 6 of the similar class of a given model by the pull-in voltage

of the lumped model to obtain the static or dynamic pull-in voltage of the model.

5 Conclusion

In this paper, the static and dynamic responses of electrostatically-actuated fixed–

fixed and cantilever microbeams to DC and step DC voltages were studied. The

governing static equation was solved by the Galerkin-based SSLM method and in

the dynamic analysis, the response of the microbeams to a step DC voltage was

obtained using the Galerkin-based reduced order model. Calculated static and

Fig. 6 The variation of the damping corrective coefficient versus damping ratio

Table 6 Comparison of results of the proposed formula with numerical analysis for a fixed–fixed and

cantilever microbeam

Static pull-in voltage Dynamic pull-in voltage

Numerical analysis Proposed

formula

Numerical analysis Proposed

formula[25] [26] [25] [26]

Fixed–fixed 4.80 – 4.80 4.40 – 4.35

Cantilever – 0.65 0.66 – 0.60 0.60

Sens Imaging (2011) 12:117–131 129

123

dynamic pull-in voltages by numerical analysis were validated by previous works

and a good agreement was achieved.

Finally, by introducing the novel dimensionless design corrective coefficients,

closed form formulas were developed for the static and dynamic pull-in analysis in

terms of static lumped pull-in voltage. The introduced design corrective coefficients

capture the effects of residual stresses, stretching and damping ratio. The results of

the proposed formulas were compared with existing results and a good agreement

was achieved. The proposed closed form formulas can be useful in the design and

analysis of similar classes of problems.

References

1. Sallese, J. M., Grabinski, W., Meyer, V., Bassin, C., & Fazan, P. (2001). Electrical modeling of a

pressure sensor MOSFET. Sensors and Actuators. A, Physical, 94, 53–58.

2. Nabian, A., Rezazadeh, G., Haddad-derafshi, M., & Tahmasebi, A. (2008). Mechanical behavior of a

circular micro plate subjected to uniform hydrostatic and non-uniform electrostatic pressure. Journalof Microsystem Technology, 14, 235–240.

3. Rezazadeh, G., Khatami, F., & Tahmasebi, A. (2007). Investigation of the torsion and bending effects

on static stability of electrostatic torsional micromirrors. Journal of Microsystem Technology, 13,

715–722.

4. Saif, M. T. A., Alaca, B. E., & Sehitoglu, H. (1999). Analytical modeling of electrostatic membrane

actuator micro pumps. Journal of Microelectromechanical Systems, 8, 335–345.

5. Rezazadeh, G., Tayefe-Rezaei, S., Ghesmati, J., & Tahmasebi, A. (2007). Investigation of the pull-in

phenomenon in drug delivery micropump using Galerkin method. Sensors & Transducers Journal,78, 1098–1107.

6. Soleymani, P., Sadeghian, H., Tahmasebi, A., & Rezazadeh, G. H. (2006). Pull-in instability

investigation of circular micro pump subjected to nonlinear electrostatic force. Sensors & Trans-ducers Journal, 69, 622–628.

7. Bao, M., & Wang, W. (1996). Future of microelectromechanical systems (MEMS). Sensors andActuators. A, Physical, 56, 135–141.

8. Senturia, S. (2001). Microsystem design. Norwell, MA: Kluwer.

9. Rezazadeh, G. (2007). A comprehensive model to study nonlinear behaviour of multilayered micro

beam switches. Microsystem Technologies, 14, 35–141.

10. Sadeghian, H., Rezazadeh, G. H., & Osterberg P. M. (2007). Application of the generalized dif-

ferential quadrature method to the study of pull-in phenomena of MEMS switches. Journal ofMicroelectromechanical Systems, 16(6), 1334–1340.

11. Osterberg, P. M., & Senturia, S. D. (1997). M-TEST: A test chip for MEMS material property

measurement using electrostatically actuated test structures. Journal of MicroelectromechanicalSystems, 6, 107–118.

12. Abdel-Rahman, E. M., Younis, M. I., & Nayfeh, A. H. (2002). Characterization of the mechanical

behavior of an electrically actuated microbeam. Journal of Micromechanics and Microengineering,12, 759–766.

13. Younis, M. I., Abdel-Rahman, E. M., & Nayfeh, A. (2003). A reduced-order model for electrically

actuated microbeam-based MEMS. Journal of Microelectromechanical Systems, 12(5), 672–680.

14. Younis, M. I., Miles, R., & Jordy, D. (2006). Investigation of the response of microstructures under

the combined effect of mechanical shock and electrostatic forces. Journal of Micromechanics andMicroengineering, 16, 2463–2474.

15. Thomson, W. T., & Dahleh, M. D. (1998). Theory of vibration with applications. USA: Prentice Hall.

16. Shigley, J. E. (1986). Mechanical engineering design. USA: McGraw-Hill.

17. Mukherjee, T., Fedder, G. K., & White, J. (2000). Emerging simulation approaches for microma-

chined devices. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems,19, 1572–1589.

130 Sens Imaging (2011) 12:117–131

123

18. Senturia, S. D., Aluru, N., & White, J. (1997). Simulating the behavior of MEMS devices. Computingin Science & Engineering, 4, 30–43.

19. Gupta, R. K. (1997). Electrostatic pull-in test structure design for in situ mechanical property

measurement of microelectromechanical systems (MEMS), Ph.D. dissertation, MIT, Cambridge,

MA, pp. 10–27.

20. Nayfeh, A., Younis, M. I., & Abdel-Rahman, E. M. (2005). Reduced-order models for MEMS

applications. Journal of Nonlinear Dynamics, 41, 211–236.

21. Lynch, J. P., Partridge, A., Law, K. H., Kenny, T. W., Kiremidjian, A. S., & Carryer, E. (2003).

Design of piezoresistive MEMS-based accelerometer for integration with wireless sensing unit for

structural monitoring. Journal of Aerospace Engineering, 16(3), 108–114.

22. Rezazadeh, G., Tahmasebi, A., & Zubtsov, M. (2006). Application of piezoelectric layers in elec-

trostatic MEM actuators: Controlling of pull-in voltage. Journal of Microsystem Technology, 12,

1163–1170.

23. Nayfeh, H., & Mook, D. T. (1979). Nonlinear oscillations. Wiley: New York.

24. Rezazadeh, G., & Tahmasebi, A. (2006). Eliminating of the residual stresses effect in the fixed–fixed

end type MEMS switches by piezoelectric layers. Sensors & Transducers Journal, 66, 534–542.

25. Osterberg P. (1995). Electrostatically actuated microelectromechanical test structures for material

property measurement, Ph.D. thesis, MIT, Cambridge.

26. Hung, E. S., & Senturia, S. D. (1999). Generating efficient dynamical models for microelectrome-

chanical systems from a few finite-element simulation runs. Journal of MicroelectromechanicalSystems, 8, 280–289.

27. Ananthasuresh, G. K., Gupta R. K., & Senturia, S. D. (1996) An approach to macromodeling of

MEMS for nonlinear dynamic simulation. In Proceeding ASME international conference of

mechanical engineering congress and exposition (MEMS), Atlanta, GA, (pp. 401–407).

28. Krylov,S., & Maimon, R. (2003). Pull-in dynamics of an elastic beam actuated by distributed

electrostatic force. In Proceeding 19th biennial conference in mechanical vibration and noise (VIB),

Chicago, IL. DETC2003/VIB-48518.

29. Nayfeh, A., Younis, M. I., & Abdel-Rahman, E. M. (2007). Dynamic pull-in phenomenon in MEMS

resonator. Journal of Nonlinear Dynamics, 48, 153–163.

Sens Imaging (2011) 12:117–131 131

123