Resonant micro beam

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Nonlinear Dynamics 31: 91117, 2003.

2003 Kluwer Academic Publishers. Printed in the Netherlands.

A Study of the Nonlinear Response of a Resonant Microbeam to

an Electric Actuation

M. I. YOUNIS and A. H. NAYFEHDepartment of Engineering Science and Mechanics, MC 0219, Virginia Polytechnic Institute and State

University, Blacksburg, VA 24061, U.S.A.; E-mail: [email protected]

(Received: 21 February 2002; accepted: 1 October 2002)

Abstract. An investigation into the response of a resonant microbeam to an electric actuation is presented. A non-

linear model is used to account for the mid-plane stretching, a DC electrostatic force, and an AC harmonic force.

Design parameters are included in the model by lumping them into nondimensional parameters. A perturbation

method, the method of multiple scales, is used to obtain two first-order nonlinear ordinary-differential equations

that describe the modulation of the amplitude and phase of the response and its stability. The model and the results

obtained by the perturbation analysis are validated by comparing them with published experimental results. Thecase of three-to-one internal resonance is treated.

The effect of the design parameters on the dynamic responses is discussed. The results show that increasing

the axial force improves the linear characteristics of the resonance frequency and decreases the undesirable fre-

quency shift produced by the nonlinearities. In contrast, increasing the mid-plane stretching has the reverse effect.

Moreover, the DC electrostatic load is found to affect the qualitative and quantitative nature of the frequency-

response curves, resulting in either a softening or a hardening behavior. The results also show that an inaccurate

representation of the system nonlinearities may lead to an erroneous prediction of the frequency response.

Keywords: MEMS, resonator, primary resonance, forced vibration.

1. Introduction

Electrically actuated microbeams have been widely used and studied by the MEMS com-

munity. They form a major component in many micromechanical devices, such as capacitive

switches [1, 2], pull-in sensors [3], and resonant sensors [416]. In this paper, we study

resonant sensors using electrically actuated microbeams.

The fundamental natural frequency of a resonant microbeam is very sensitive to the axial

strain. External loads, such as pressure, temperature, force, and acceleration, apply axial

strains on a microbeam, leading to a shift in its natural frequencies. This shift is readily

converted to a digital signal, which is related to the physical quantity being measured. The

fundamental natural frequency also can be affected by other factors, such as squeeze-film

damping [4] and the elasticity of the microbeam supports [5].

The electric load and the mechanical restoring force govern the behavior of a microbeam.The electric load is composed of a DC polarization voltage and an AC voltage. The DC

component applies an electrostatic force on the microbeam, thereby deflecting it to a new

equilibrium position, while the AC component vibrates the microbeam around this equilibrium

position. The combined electric load has an upper limit beyond which the mechanical restoring

force can no longer resist its opposing force, thereby leading to the collapse of the microbeam.

This structural instability phenomenon is known as pull-in, and the critical voltage associated


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92 M. I. Younis and A. H. Nayfeh

with it is called pull-in voltage. Younis et al. [6] showed that the pull-in voltage corresponds

to a saddle-node bifurcation. They calculated the static deflection at each DC voltage and

found two solution branches: the larger solution is unstable and the smaller one is stable. As

the DC voltage increases, these branches get closer to each other, coalesce , and destroy each

other in a saddle-node bifurcation, corresponding to the pull-in voltage. The deflection at pull-

in is approximately 57% of the airgap space [6, 7], which separates the microbeam from the

stationary substrate.

A microbeam restoring force is composed of three components: bending, axial force, and

mid-plane stretching due to immovable boundaries. The mechanical restoring force tends to

shift the natural frequencies to higher values, while the electrostatic force tends to shift the

natural frequencies to smaller values.

A number of studies have investigated the problem of a microbeam that is initially deflected

by a DC electrostatic force and driven by an AC force. Zook et al. [8] reported experiment-

ally that increasing the driving AC voltage leads to an increase in the resonance frequency

(hardening behavior). Also, they observed a hysteresis that depends on the direction of the

frequency sweep.

Tilmans and Legtenberg [9] approximated the nonlinear resonance frequency as the fun-

damental natural frequency for small values of the AC forcing amplitude. For large-amplitudeAC excitations, they approximated the dynamic problem using Rayleighs energy method,

modified to account for electric forces and mid-plane stretching. They derived an equation that

governs the resonance frequency as a function of the excitation amplitude. They compared the

results obtained using this equation to their experimental results. Although they found a good

qualitative agreement, showing a hardening effect, the agreement was poor quantitatively.

They concluded that such a hardening behavior is more severe for high-quality factors, high

DC voltages, and high AC voltages, whereas large axial strains can decrease it.

Gui et al. [10] investigated the dynamic problem using Rayleighs energy method, modi-

fied to account for electric forces and mid-plane stretching. They neglected the applied axial

force. They derived an equation for the fundamental resonance frequency, which predicts a

hardening behavior. They also derived a criterion that defines a region of DC and AC voltages

for a hysteresis-free operation. They compared their theoretical and experimental results and

found good agreement.

Turner and Andrews [11] approximated the nonlinear resonance frequency of a microbeam

using a perturbation method. The problem was modeled using a spring-mass system with a

cubic nonlinearity representing mid-plane stretching. Using the method of harmonic balance,

they derived equations describing the microbeam resonance frequency for two separate cases.

In one case, they neglected the electrostatic force and included mid-plane stretching, and,

in the second case, they included the electrostatic force and neglected mid-plane stretching.

Using the equations obtained from both cases, they derived an equation that compensates for

the increase in the resonance frequency due to mid-plane stretching and the decrease in the

resonance frequency caused by the electrostatic force. The results showed that eliminating the

effect of the frequency dependence on the amplitude requires a very high DC voltage, whichmay not be attainable.

Ayela and Fournier [12] experimentally studied the response of microbeams of different

geometric shapes to a general electric load composed of DC and AC components. They

made several plots to show variation of the resonance frequency with the excitation amplitude

for different axial loads. The experimental data were used to extract parameters that fit the

equation of a spring-mass system with cubic nonlinearity and a linear electric force (i.e., the


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Nonlinear Response of Resonant Microbeams to Electric Actuations 93

electric force is assumed to be independent of the gap distance). The experimental results

showed that, for the operating conditions used, some devices exhibited softening behavior,

whereas others exhibited hardening behavior. They concluded that a nonlinear behavior may

result from many distinct phenomena and that each case has to be studied separately. They

added that the nonlinear behavior of these devices may be investigated experimentally, but

there is no way of predicting it analytically.

Veijola et al. [13] modeled a microbeam using a spring-mass model with a cubic nonlin-

earity representing mid-plane stretching. They included the effect of the driving AC voltages,

but neglected the effect of the DC polarization voltages. They used the method of harmonic

balance to show that the nonlinearity of the electrostatic forces can cause a softening-type

behavior, whereas the mid-plane stretching can cause a hardening-type behavior. They con-

ducted an experiment, which showed a hardening behavior. They used the results to extract

values for the parameters involved in their model. They compared the experimental results

with those obtained using their model and found them to be in good agreement. However,

they did not give an explanation for observing a hardening rather than a softening behavior.

From the aforementioned overview, we note that the literature lacks a comprehensive

model that predicts the dynamic response of a microbeam to an electric excitation over general

operating conditions. Most existing models are formulated with a pre-assumed behavior basedon experimental observations, which limits the analysis to specific design parameters only.

For instance, the nonlinearity of the system is assumed to be cubic and positive to justify

the observed hardening behavior whereas the nonlinearity of the electric forces, which tends

to yield softening behavior, is ignored. In some models, the system nonlinearity is totally

ignored, thereby restricting the analysis to small values of AC voltages and DC polarization

voltages. Further, in many cases, the microbeam is modeled as a single-degree-of-freedom

system, thereby neglecting its mass distribution. The electric force is included incompletely

in some papers by neglecting one of its components. In other cases, the dependence of the

electric forces on the variable gap is ignored. In addition, most of the previous works do not

recognize the fact that a deflected microbeam caused by DC polarization voltages has natural

frequencies different from those of a straight beam; thus, the models are valid only for very

low DC polarization voltages.

In a previous work [7], we introduced a nonlinear model for a microbeam, which accounts

for the initial deflection due to a DC electrostatic force, mid-plane stretching, and an applied

axial load. The model does not account for the effects of the shear deformation and rotary

inertia. We analyzed the static deflection of the microbeam and investigated its linear natural

frequencies and mode shapes when actuated by a DC electrostatic force. Here, we expand on

that work by considering the microbeam response to a general electric load composed of both

DC and AC components. We model the microbeam as a distributed-parameter system. The

equations are nondimensionalized and the design parameters of the resonator are lumped into

nondimensional parameters, which are then used to develop a general equation that describes

the dynamics of the microbeam. We apply a perturbation method, the method of multiple

scales, to the general equation of motion and its associated boundary conditions, to obtainan approximation of the response of the microbeam to a primary-resonance excitation. We

derive equations that describe the nonlinear resonance frequency, the amplitudes of periodic

solutions, and the stability of these solutions. We study the effect of the design parameters on

the nonlinear resonance frequency and the effective nonlinearity of the system. We compare

the theoretical results with available experimental data. Then, we investigate the possibility

of activating a three-to-one internal resonance between the first and second modes, which is


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Figure 1. A schematic drawing of a resonant microbeam.

the most likely case to occur during the normal operation of the resonator. The absence of

nonlinear interactions is necessary for a stable and reliable device.

2. Problem Formulation

We consider a resonant microbeam (Figure 1) actuated by an electric load composed of a

DC component (polarization voltage) Vp and an AC component v(t) and subject to a viscous

damping c per unit length.

We model the microbeam as a plate undergoing cylindrical bending under an applied load,which is constant along the width of the plate. Typical dimensions of resonators violate the

basic assumptions of a beam model; the ratio of a typical microbeam length to width b is

< 10. The plate is clamped across its width and free across its long ends. We assume that the

transverse deflection w(x,t) of the plate is constant along the width, thus reducing the plate

static deflection equation to [17]

EI

1 2 4w

x4=

EA

2(1 2)

0

w

x

2dx + N

2w

x2, (1)

where x is the position along the plate length, A and I are the area and moment of inertia

of the cross section (A = bh and I = (1/12)bh

3

, where h is the microbeam thickness), Eis Youngs modulus, is Poissons ratio, and N is the applied tensile axial force. The plate

boundary conditions are

w(0, y) = w(, y) = 0,w(0, y)

x=

w(,y)

x= 0. (2)

Using the modified beam equation, we rewrite the equation of motion governing the transverse

deflection of the microbeam under electric forces as

EI

1 24w

x4+ A

2w

t2+ c

w

t=

EA

2(1 2)

0

w

x

2dx + N

2w

x2

+1

20r b

Vp + v(t)

2

(d w)2, (3)

where w(x,t) is the deflection of the microbeam, t is time, is the material density, d is the

gap width, 0 is the dielectric constant of vacuum, and r is the relative dielectric constant of

the gap medium to that of an air gap. The last term in Equation (3) represents the parallel-

plate electric forces [18] assuming a complete overlapping area between the microbeam and


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the stationary electrode. The microbeam is assumed to be clamped at both ends; hence the

boundary conditions are

w(0, t) = w(,t) = 0,w(0, t)

x=

w(,t)

x= 0. (4)

For convenience, we introduce the nondimensional variables (denoted by stars)

w =w

d, x =

x

, t =

t

T, (5)

where T is a time scale, which is defined below. Substituting Equations (5) into Equations (3)

and (4) and dropping the stars, we obtain

4w

x4+

2w

t2+ c

w

t= [1(w,w) + N]

2w

x2+ 2

(Vp + v(t))2

(1 w)2, (6a)

w(0, t) = w(1, t) = 0,w(0, t)

x=

w(1, t)

x= 0, (6b)

where

(f1(x,t),f2(x,t)) =

10

f1

x

f2

xdx. (6c)

Equation (6a) is a nondimensional integral-partial-differential equation with linear and non-

linear terms as well as external and parametric excitation terms. The parameters appearing in

Equation (6a) are

c =c4(1 2)

EI T

, 1 = 6dh

2

, N =N 2(1 2)

EI

, 2 =60r

4(1 2)

Eh3d3(7)

and T is chosen as

T =

bh4(1 2)

EI

1/2.

Typically, T is in the ranges of microseconds. The ranges of the rest of the nondimensional

parameters are c 100300, 1 0.110, N 10100, and 2V2

p 0.150.

The microbeam deflection under an electric force is composed of a static component due to

the DC voltage, denoted by ws (x), and a dynamic component due to the AC voltage, denoted

by u(x,t); that is,

w(x,t) = ws (x) + u(x, t). (8)

To calculate the static deflection of the microbeam, we set the time derivatives and the AC

forcing term in Equation (6a) equal to zero and obtain

d4ws

dx4= [1(ws , ws ) + N]

d2ws

dx2+

2V2

p

(1 ws )2

. (9a)


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It follows from Equations (6b) and (8) that the boundary conditions are

ws = 0 anddws

dx= 0 at x = 0 and x = 1. (9b)

We generate the problem governing the dynamic behavior of the microbeam around the

deflected shape by substituting Equation (8) into Equations (6) and using Equations (9) to

eliminate the terms representing the equilibrium position. To third-order in u, the result is 2u

t2+ c

u

t+

4u

x4= (1(ws , ws ) + N )

2u

x2+ 21(ws , u)

d2ws

dx2

+22V

2p

(1 ws )3

u + 1(u, u)d2ws

dx2+ 21(ws , u)

2u

x2

+32V

2p

(1 ws )4u2 + 1(u, u)

2u

x2+

42V2

p

(1 ws )5u3

+22Vp

(1 ws )2v(t) +

42Vp

(1 ws )3v(t)u + , (10a)

u(0, t) = u(1, t) = 0, u(0, t)x

= u(1, t)x

= 0, (10b)

where the term involving v2(t) is dropped because typically v2(t) V2p in resonant sensors.

We end this section with two notes regarding the choice of the boundary conditions and the

use of the linear damping in the model. First, the fixed-fixed end conditions are idealization

to actual cases which may have some elasticity [5]. Some structures employ step-up type

supports, which are not perfectly rigidly fixed [19]. According to [19], the flexibility of the

supports reduces the critical value of the buckling load to be less than that for ideal fixed-

fixed ends. Bouwstra and Geijselaers [5] mentioned that this flexibility makes the resonance

frequencies lower than the predicted values for ideal fixed-fixed ends. The second note relates

to the so-called squeeze-film damping. When a resonator moves too close to a stationary

substrate, additional damping forces, the squeeze-film damping, increases due to the pressurebuilt up in the airgap [14]. Because typically resonant sensors are placed in a vacuum cavity in

low pressure under 0.2 mbar, the resonance frequencies are almost independent of the pressure

change [9, 15].

3. Linear Mode Shapes and Corresponding Natural Frequencies

We drop the nonlinear, forcing, and damping terms in Equation (10a) and obtain the linear

undamped eigenvalue problem

4u

x4 (11 + N )

2u

x2

22V2

p

(1 ws )3u 21(ws , u)

d2ws

dx2=

2u

t2, (11a)

u = 0 andu

x= 0 at x = 0 and x = 1, (11b)

where 1 = (ws , ws ). We solve Equations (11) for the undamped mode shapes and natural

frequencies under the static deflection distribution ws (x). Assuming a harmonic motion in the

nth mode shape

u(x,t) = n(x) ein t, (12)


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where n(x) is the nth mode shape and n is the nth nondimensional natural frequency, we

reduce Equations (11) to

ivn (11 + N )n

22V2

p

(1 ws )3n 212w

s =

2nn, (13a)

n = n = 0 at x = 0 and x = 1, (13b)

where 2 = (ws , n) and the prime denotes the derivative with respect to x.

4. Forced Vibration: Single-Mode Approximation

4.1. PERTURBATION ANALYSIS

We investigate the nonlinear vibrations of a clamped-clamped microbeam subject to an electric

load composed of DC and AC components. We analyze its nonlinear response to a primary-

resonance excitation of its first mode because it is the case that is used in resonator applica-

tions. Still, the analysis is general and can be used to study nonlinear responses of the beam

to a primary resonance of any of its mode.

There are at least two approaches for determining the response of a distributed-parameter

system with quadratic and cubic nonlinearities. In the first approach, called discretization

approach, one discretizes the governing integral-partial-differential equation and associated

boundary conditions, keeps enough modes in the discretization, and uses a perturbation method

to attack the discretized system [20]. In the second approach, called the direct approach, one

attacks directly the distributed-parameter problem. In this paper, we attack Equations (10) dir-

ectly using the method of multiple scales [20, 21] to determine a uniformly valid approximate

solution. To this end, we seek a second-order uniform solution of Equations (10) in the form

u(x,t; ) = u1(x,T0, T2) + 2

u2(x,T0, T2) + 3

u3(x,T0, T2) + , (14)

where is a small nondimensional bookkeeping parameter, T0 = t, and T2 = 2t. We note

that u(x,t) does not depend on T1 = t because, as shown below, secular terms arise at

O( 3). In order that the nonlinearity balances the effects of the damping c and excitation v(t),

we scale them so that they appear together in the modulation equations as follows: 2c and

3VAC cos(t) where VAC is the magnitude of the applied AC voltage and is the excitation

frequency. Substituting Equation (14) into Equation (10a) and equating coefficients of like

powers of, we obtain

order

L(u1) = D20 u1 + u

iv1 11u

1 N u

1 21(ws , u1)w

s

22V2p

(1 ws )3u1 = 0; (15)

order 2

L(u2) = 1ws (u1, u1) + 21(ws , u1)u

1 +

32V2

p

(1 ws )4

u21; (16)


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order 3

L(u3) = 2D0D2u1 cD0u1 + 21ws (u1, u2) + 21(ws , u2)u

1 + 21(ws , u1)u

2

+ 1(u1, u1)u1 + 2F(x) cos(T0) +

62V2

p

(1 ws )4u1u2 +

42V2

p

(1 ws )5u31, (17)

where

F(x) =2VpVAC

(1 ws )2.

The boundary conditions at all orders are given by

uj = 0 and uj = 0 at x = 0 and x = 1, j = 1, 2, 3. (18)

The first-order problem given by Equations (15) and (18) is identical to the linear ei-

genvalue problem, Equations (11). Because in the presence of damping the homogeneous

solutions corresponding to all modes that are not directly or indirectly excited decay with

time [21], the solution of Equations (15) and (18) is assumed to consist of only the directly

excited mode. Accordingly, we express u1

as

u1 = A(T2) eiT0 (x) + A(T2) e

iT0 (x), (19)

where A(T2) is a complex-valued function that is determined by imposing the solvability

condition at third order, the overbar denotes the complex conjugate, and and (x) are the

natural frequency and corresponding eigenfunction of the considered mode, respectively. The

eigenfunction (x) is normalized such that1

02 dx = 1.

Substituting Equation (19) into Equation (16), we obtain

L(u2) = (A2 e2iT0 + 2AA + A2 e2iT0 )h(x), (20)

where

h(x) = 1(,)ws + 21

(ws , ) + 32V

2

p

(1 ws )42.

The solution of Equations (20) and (18) can be expressed as

u2 = 1(x)A2 e2iT0 + 22(x)AA + 1(x)A

2 e2iT0 , (21)

where 1 and 2 are the solutions of the boundary-value problems

M(i , 21i ) = h(x), (22)

j = 0 and j = 0 at x = 0 and x = 1, j = 1, 2 (23)

and ij is the Kronecker delta and the linear differential operator M(,) is defined as

M(,) = iv 2 + 11 N 21(ws ,)w

s

22V2

p

(1 ws )3. (24)

To describe the nearness of the excitation frequency to the fundamental natural fre-

quency , we introduce the detuning parameter defined by

= + 2. (25)


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Substituting Equations (19), (21), and (25) into Equation (17) and keeping the terms that

produce secular terms only, we obtain

L(u3) =

i (2A + cA)(x) + (x)A2A + F(x) ei T2

eiT0 + cc + NST, (26)

where A denotes the derivative of A with respect to T2, cc denotes the complex conjugate of

the preceding terms, and NST stands for terms that do not produce secular terms. The function

(x) is defined as

(x) = Gq (x) + Gc (x) +

Eq (x) +

Ec (x),

where

Gq (x) = 21ws (1, ) + 41w

s (2, ) +

21

1 + 41

2

(ws , )

+ 21(ws , 1) + 41(ws , 2),

Gc (x) = 31(,),

Eq (x) =62V

2p

(1 ws )4

(22 + 1) ,

Ec (x) =42V

2p

(1 ws )53.

The subscripts q and c denote quadratic and cubic nonlinear terms and the superscripts G and

E denote terms produced by the geometric and electric nonlinearities.

For a uniform second-order approximation, we do not need to solve for u3. Only, we need to

impose the solvability condition, which provides an equation for the function A. Because the

homogeneous problem governing u3 has a nontrivial solution, the corresponding nonhomo-

geneous problem has a solution only if the right-hand side of Equation (26) is orthogonal

to every solution of the adjoint homogeneous problem governing u3 [20]. We note that our

problem is self adjoint, the adjoints are given by (x) eiT0 . Multiplying the right-hand side

of Equation (26) with (x) eiT0 and integrating the result from x = 0 to x = 1, we obtain

the following solvability condition:

2i(D2A + A) + 8SA2A F ei T2 = 0, (27)

where

S = SGq + SGc + S

Eq + S

Ec ,

SGq = 1

8

10

Gq dx,

SGc = 1

8

10

Gc dx,

SEq = 1

8

10

Eq dx,


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SEc = 1

8

10

Ec dx,

=1

2

1

0

c2 dx,

F =

10

F dx.

Here, S is the effective nonlinear coefficient of the system and Smn denotes a nonlinear coeffi-

cient of source m (electric or geometric) and of order n (quadratic or cubic). Ifc is independent

ofx, then = c/2. In general, c is a function ofx.

Next, we express A in the polar form A = (1/2)a ei , where a and are real-valuedfunctions, representing, respectively, the amplitude and phase of the response. Substituting

for A in Equation (27), separating the real and imaginary parts, and letting = T2 , we

obtain the following modulation equations:

a = a +F

sin , (28)

a = a Sa3

+

F

cos . (29)

Substituting Equations (19) and (21) into Equation (14) and setting = 1, we express, to the

second approximation, the microbeam response to the external excitation as

u(x,t) = a cos(t )(x) +1

2 a2

2(x) + cos2(t )1(x)

+ , (30)

where a and are governed by Equations (28) and (29).

It follows from Equation (30) that periodic solutions of Equations (10) correspond to

constant a and ; that is, the fixed points (a0, 0) of Equations (28) and (29). Thus, letting

a = = 0 in Equations (28) and (29) and eliminating 0 from the resulting equations, we

obtain the following frequency-response equation:

a20

2 +

Sa 20

2=

F2

2(31)

Equation (31) is an implicit equation for the amplitude a of the periodic response as a func-tion of the detuning parameter (which is a representation of the excitation frequency), the

effective nonlinear coefficient S, the damping coefficient , and the amplitude of excitation

F. Solving Equation (31) for , we obtain

= 1

a0

F2

2 2a20

1/2+

Sa 20

. (32)


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Recalling that = 2, setting = 1, and noting that the amplitude a is maximum

when the expression inside the square root vanishes, we obtain the following equation for the

nonlinear resonance frequency:

r = +SF2

32. (33)

The parameters is related to the quality factor Q by

=

2Q(34)

Substituting Equation (34) into Equation (33), we obtain

r = +4SQ2

5F2. (35)

Equation (35) relates the nonlinear resonance frequency r to the effective nonlinearity S

of the system, the amplitude F of the AC forcing, and the quality factor Q. Equation (35) can

be used to predict the resonance frequency for excitation frequencies much less than twice the

fundamental natural frequency. This because when 2 a principal parametric resonanceand a subharmonic resonance of order one-half will be activated.

The stability of the fixed points (a0, 0) is determined by examining the eigenvalues of the

Jacobian matrix of Equations (28) and (29) evaluated at the corresponding fixed point [22].

The characteristic equation is

2 + 2 +

2 +

3Sa 20

Sa 20

= 0. (36)

For asymptotically stable solutions, all of the eigenvalues must be in the left-half plane.

4.2. RESULTS

To describe the dynamic response of the microbeam, we need to determine the natural fre-

quency , the excitation amplitude F, the effective nonlinearity of the system S, and the

damping coefficients or Q.

As a first step, we solved the boundary-value problem, Equations (9), for the deflection

due to the DC electrostatic force. We did this iteratively using a shooting method to de-

termine the value of the integral (ws , ws ). Using the converged solution ws , we solved

the boundary value problem, Equations (13), for the fundamental natural frequency and its

corresponding eigenfunction. We used a shooting method to iterate on 2, , and until they

converged to within a predefined tolerance. Next, we solved the boundary-value problems,

Equations (2224) to evaluate the functions 1 and 2 using a method similar to that used to

solve Equations (13). Finally, we evaluated the s, Ss, and F.

We begin by comparing the results obtained by using the present model with the experi-mental results of Tilmans and Legtenburg [9] and Gui et al. [10]. Results from other papers,

which are mentioned in the introduction, are not used because either they were produced from

another device configuration [11, 12], for which our model does not apply, or the authors did

not give enough data about the specifications of their devices [8, 13].

In Figure 2, we compare the normalized nonlinear resonance frequency r / obtained

using Equation (35) with those obtained theoretically and experimentally by Tilmans and


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Figure 2. Comparison of the normalized nonlinear resonance frequency r / calculated using our model (solid

lines) with those obtained theoretically (dashed lines) and experimentally by Tilmans and Legtenburg [9] for two

microbeams of lengths 210 m (diamonds) and 310 m (circles). The thin solid lines are based on the quality

factors of Tilmans and Legtenburg [9] and the thick solid lines are based on quality factors estimated using our

model.

Legtenburg [9] for two microbeams of lengths 210 and 310 m with h = 1.5 m, b =

100 m, d = 1.18 m, and subject to an axial load of 0.0009 Newton. The reported pull-

in voltages for the 210 and 310 m microbeams are 28 and 13.8 V, respectively. The DC

polarization voltage Vp = 1 V for data points of the first microbeam except for the first two

points where Vp = 2 V. For the second microbeam, Vp = 2 V for all data points. The theory

of Tilmans and Legtenburg [9] is based on Rayleighs energy method modified to account for

electric forces and mid-plane stretching. The latter effect however was misrepresented, as we

show in Appendix A.

We show two sets of calculated results for each microbeam. The first set, shown as thin

solid lines, uses the same values of Q reported by Tilmans and Legtenburg [9]. They extracted

the values of Q analytically using an equation derived from their model. The parameters

of this model were obtained by fitting the predicted frequency-response curve at a low DC

voltage to the one obtained experimentally. These quality factors are Q = 592 and 151 for

the 210 and 310 m microbeams, respectively. As shown in Figure 2, these values give poor

results. Legtenburg and Tilmans [16] mentioned that the quality factor for the design of their

device varies across the wafer due to variations in the sealing pressure of the microbeam

encapsulation. Because such variations can lead to a wrong measurement ofQ and due to thedifficulty of measuring the system damping, in general, we determined the quality factors by

matching r / obtained using Equation (35) at VAC = 0.6 V to the experimental value of

Tilmans and Legtenburg [9].

We obtained Q = 816.6 and 197 for the 210 and 310 m microbeams, respectively. The

results obtained using these values, shown as thick solid lines, are in excellent agreement with

the experimental results.


13/27


Figure 3. Comparison of the hysteretic points obtained using our model (solid lines) with those obtained theoret-

ically (dashed line) and experimentally (circles) by Gui et al. [10]. The thin solid line is based on the quality factor

Q = 900 reported by Gui et al. [10] and the thick solid line is based on a quality factor Q = 1000 estimated using

our model.

In Figure 3, we show another comparison for the hysteretic points (points correspond to

the largest AC and DC voltages below which the response is single-valued and above which it

is multi-valued) predicted by our model with those obtained theoretically and experimentally

by Gui et al. [10] for a microbeam of length 210 m and h = 1.5 m, b = 100 m, d =

1 m, and Q = 900. The theoretical results of Gui et al. [10] were obtained using a modified

Rayleighs energy method that accounts for mid-plane stretching and electric forces. Mid-

plane stretching however was misrepresented, as we show in Appendix A.

In Figure 3, we show also two curves obtained using our theory corresponding to two dif-

ferent values of Q: the thick solid line was obtained using Q = 1000, which was determined

by matching our result at Vp = 4 V to the experimental result of Gui et al. [10]. The thin solid

line was obtained using Q = 900, which is the value used by Gui et al. [10]. We estimated the

axial force to be N = 0.00011 Newton by fitting the theoretically obtained natural frequency

to the experimental value obtained by Gui et al. [10]. There is excellent agreement between

our results and the experimental results.

Figures 4 and 5 show the frequency-response curves (amplitude and phase responses)corresponding to Vp = 8 V in Figure 3. In both figures, thin lines are for VAC = 0.007 V

and thick lines are for VAC = 0.03 V. The unstable regions in the latter case are shown as

dashed lines. The results are obtained by solving Equations (31) and (36).

Next, we study the effect of the nondimensional design parameters on the nonlinear reson-

ance frequency. As indicated by Equation (35), the nondimensional design parameters affect

the nonlinear resonance frequency in two ways: changing the effective nonlinearity of the


14/27


Figure 4. The amplitude of the dynamic response versus the frequency of excitation corresponding to Vp = 8 V

in Figure 3.

Figure 5. The phase of the dynamic response versus the frequency of excitation corresponding to Vp = 8 V in

Figure 3.


15/27


Figure 6. Variation of the normalized nonlinear resonance frequency r / with VAC for various values of the

nondimensional axial load N. The values of1, 2V2

p , and Q are 3.7, 5.5, and 796, respectively.

system S and the fundamental natural frequency . We refer to Younis et al. [7] for the pull-in

voltages of the numerical data used in the following figures.

In Figure 6, we show the effect of varying the driving voltage AC on r / for various val-

ues of the nondimensional axial load N. Increasing the driving voltage amplitude AC increase

r /. On the other hand, increasing the axial force N at a constant VAC decreases r / and

also increases its linear behavior. We note that negative values of N, which correspond to

compressive loads, have a greater effect on r /.

In resonant sensors, residual stresses are typically of the tensile type. They are inducedduring the fabrication process. Legtenberg and Tilmans [16] mentioned that such stresses are

desirable to reduce the possibility of buckling. However, because resonant sensors may exhibit

temperature variations, which may induce compressive residual stresses, operation away from

the critical buckling load should be ensured. As an example, we calculated a critical buckling

load Pcr = 13.3037 for the microbeam in Figure 2 of length 210 m using the formula

Pcr =4 Eh3

12L2(1 2)

for a fixed-fixed beam.

In Figure 7, we show the effect of varying the driving voltage amplitude AC on r / for

various values of 1. It can be seen that, for values of 1 greater than 0.05, increasing ACincreases r /. In contrast, for values of1 less than 0.05, increasing AC leads to a decrease

in r /. We also note that increasing 1 increases the nonlinear behavior of r / and its

value at a constant value of AC.

In Figure 8, we show the effect of varying AC on r / for various values of 2V2

p . In-

creasing 2V2

p at a constant VAC increases r / up to a value of 2V2

p 35. Above this

value, the increase in r / becomes smaller until it is reversed completely at 2V2

p 41,


16/27


Figure 7. Variation of the normalized nonlinear resonance frequency r / with VAC for various values of 1.

The values ofN, 2V2

p , and Q are 8.7, 5.5, and 796, respectively.

Figure 8. Variation of the normalized nonlinear resonance frequency r / with VAC for various values of2V2

p .

The values of1, N, and Q are 3.7, 8.7, and 796, respectively.


17/27


Figure 9. Variation of the nonlinear coefficients with the axial load N. The values of 1 and 2V2

p are 3.7 and

5.5, respectively.

where the nonlinear resonance frequency becomes highly sensitive to changes in the DC

polarization voltage. In fact, a slight increase in 2V2

p beyond this value leads to a sharp drop,

which can even reach zero, in the nonlinear resonance frequency with increasing VAC. We

note that such a strange behavior occurs before reaching the pull-in limit, which in this case

occurs at 2V2

p 90. This result shows that the nondimensional parameter 2V2

p changes

the dynamic response of the microbeam from a hardening to a softening response. This is

because, for this range of2V2

p , the electrostatic force, which tends to lower the resonance fre-

quency, drastically dominates the mid-plane stretching, which tends to increase the resonancefrequency.

4.3. DISCUSSION

In order to better understand how the nondimensional parameters N, 1, and 2V2

p affect

the effective nonlinearity of the system, we study the influence of each parameter on the

nonlinear coefficients SGq , SGc , S

Eq , S

Ec , and S. In Figure 9, we show variation of these nonlinear

coefficients with increasing the axial load N. It shows that increasing N in the positive range

leads only to a slight linear decrease in the mid-plane stretching represented by SGc , and hence

in the effective nonlinear coefficient S. However, this is not the major factor that leads to the

decrease in r / with increasing N, which is shown in Figure 6. Increasing N also increases

, which according to Equation (35) decreases the shift in r /. This agrees with the resultsof Figure 6, where the ratio r / approaches unity for large N.

In Figure 10, we show variation of the nonlinear coefficients with 1. Increasing 1 in-

creases linearly the mid-plane stretching coefficient SGc , without affecting the other nonlinear

coefficients. The increase in SGc , and so in S, increases r /, as indicated by Equation (35).

On the other hand, increasing 1 increases , which tends to decrease r /. The effect of

increasing S however dominates the later effect. This explains the increase in the normalized


18/27


Figure 10. Variation of the nonlinear coefficients with 1. The values ofN and 2V2

p are 8.7 and 5.5, respectively.

nonlinear resonance frequency with increasing 1 at a constant VAC, which is seen in Figure 7.

The effective nonlinear coefficient S starts with a negative value for very small values of 1,

which explains the qualitative change observed in Figure 7 near this range of values.

In Figure 11, we show variation of the nonlinear coefficients with 2V2

p . For small values

of2V2

p , the effective nonlinear coefficient Sdecreases with increasing 2V2

p up to values near

40 where the quadratic electric term SEc starts to decrease sharply, thereby dominating all other

nonlinear coefficients. At 2V2

p

41, the sign of S changes from positive, corresponding to

a hardening behavior, to negative, corresponding to a softening behavior, which explains the

qualitative change in the dynamic behavior observed in Figure 10. This is a significant result;

it shows that using a spring-mass model with a cubic nonlinearity, which is the model usually

used in the literature, is inaccurate and might lead to wrong results. Such models neglect the

quadratic nonlinearity, which is due to the electrostatic force and the static deflection. As

clearly shown in Figure 11, this nonlinearity becomes dominant and governs the dynamic

behavior beyond a critical value of2V2

p . Further, we note that assuming a single value for the

effective nonlinearity in the model may introduce another source of error because, as shown

in Figure 11, the effective nonlinearity is a strong function of the electrostatic force.

We note that the DC voltage needed to achieve a linear behavior in the nonlinear resonance

frequency (corresponding to S = 0) is relatively high (Vp 19V for the data of Figure 2)

and may not be attainable in resonant sensor applications. This result is in agreement with thatfound by Turner and Andrews [11].

Next, we examine the sensitivity of the dynamic response of the microbeam to the quality

factor. In Figures 12 and 13, we show variation of the amplitude of the response a with the

detuning parameter for a microbeam with 1 = 3.7 and N = 8.7 and quality factors ofQ =

200, Q = 500, and Q = 796. We use 2V2

p =15 and 45 in Figures 12 and 13, respectively.

Both figures show that increasing the quality factor amplifies the effect of the nonlinearity and


19/27


Figure 11. Variation of the nonlinear coefficients with 2V2

p . The values of1 and N are 3.7 and 8.7, respectively.

Figure 12. Variation of the response amplitude a of a microbeam with the detuning parameter for 1 = 3.7,

N = 8.7, and 2V2

p =15.


20/27


Figure 13. Variation of the response amplitude a of a microbeam with the detuning parameter for 1 = 3.7,

N = 8.7, and 2V2

p =45.

can change the dynamic state from a single-valued response, as in the Q = 200 curves, to a

multi-valued response, as in the Q = 500 and 796 curves. As can be noted from Figure 11,

the response curves exhibit a hardening behavior in Figure 12 and a softening behavior in

Figure 13.

5. Three-to-One Internal Resonance

5.1. PERTURBATION ANALYSIS

In this section, we consider modal interactions among the microbeam modes involving the

first mode. It follows from Figure 14 that 1 is away from (1/2)n for any n. Hence, a two-

to-one internal resonance between 1 and n cannot be activated. However, 1 (1/3)2 for

some range of Vp, and hence we study the possibility of activating a 1:3 internal resonance

between the first and second modes when the first mode is excited with a primary resonance.

We apply the method of multiple scales directly to Equations (10). The analysis presen-

ted here is general and applicable to any two modes whose frequencies are in the ratio of

three-to-one. We seek a solution of Equations (10) in the form of Equation (14) and obtain

Equations (1518). Because in the presence of damping all modes that are not directly or

indirectly excited decay with time [16], the solution of Equations (15) and (18) is assumed toconsist of the two interacting modes; that is,

u1 = An(T2) einT0 n(x) + Am(T2) e

imT0 m(x) + cc, (37)


21/27


Figure 14. Variation of the first four natural frequencies, calculated by Younis et al. [7], of a microbeam with

2V2

p for 1 = 3.7 and various applied axial loads.

where m and n denote the two modes being considered and the j are normalized such that10

2j dx = 1. Substituting Equation (37) into Equation (16) yields

L(u2) = A2n e

2inT0 h1n(x) + A2m e

2imT0 h1m(x) + AnAnh1n(x) + AmAmh1m(x)

+ (AnAm ei(n+m)T0 + AnAm e

i(nm)T0 )Hnm(x) + cc, (38)

where

h1i(x) = 1(i , i )ws + 21

i (ws , i ) +

32V2

p

(1 ws )42i ,

Hij(x) = 21(i , j)ws + 41

i (ws , j) +

62V2

p

(1 ws )4i j,

and the functional is defined in Equation (6c).

We express the solution of Equations (38) and (18) as

u2 = 1n(x)A2

n

e2inT0 + 1m(x)A2

m

e2imT0 + 3(x)AnAm ei(n+m)T0

+ 4(x)AnAm ei(nm)T0 + 2n(x)AnAn + 2m(x)AmAm + cc, (39)

where the j and ij are the solutions of the boundary-value problems

M(1i , 2i ) = h1i (x), (40a)

M(2i , 0) = h1i (x), (40b)


22/27


M(3, n + m) = Hnm(x), (40c)

M(4, n m) = Hnm(x). (40d)

The operator M(,) is defined in Equation (24). The boundary conditions for the j and

ij are

= 0 and = 0 at x = 0 and x = 1. (41)

Substituting Equations (37) and (39) into Equation (17) and considering the case n 3mand either n or m, we obtain

L(u3) =

in(2An + cAn)n(x) + 1n(x)A

2nAn + nm(x)AnAmAm

ein T0

+

im(2Am + cAm)m(x) + 1m(x)A

2mAm + mn(x)AmAnAn

eimT0

+ 5A3m e

3imT0 + 6AnA2m e

i(n2m)T0 + F(x) eiT0 + cc + NST, (42)

where Aj is the derivative ofAj with respect to T2 and

1i = 21(1i , i )ws + 41(2i , i )w

s + 21(ws , 1i )

i + 41(ws , 2i )

i

+ 31(i , i )i + 21(ws , i )

1i + 41(ws , i )

2i +

122V2

p

(1 ws )5

3i

+62V

2p

(1 ws )4(ii i + 2ii ),

ij = 21(3, j)ws + 21(4, j)w

s + 41(i , 2jw

s + 21(ws , 3)

j

+ 21(ws , 4)j + 41(ws , 2j)

i + 21(ws , j)

3 + 21(ws , j)

4

+ 41(ws , i )2j) + 21(j, j)

i + 41(i , j)

j +

242V2

p

(1 ws )5

i 2j

+62V

2p

(1 ws )4(3j + 4j + 22ji ),

5 = 21(1m, m)ws + 21(1m, ws )

m + 21(ws , m)

1m + 1(m, m)

m

+62V

2p

(1 ws )41mm,

6 = 21(4, m)ws + 21(1m, n)w

s + 21(4, ws )

m + 21(1m, ws )

n

+ 21(ws , m)4 + 21(n, ws )

1m + 1(m, m)

n + 21(n, m)

m

+62V

2p

(1 ws )44m +

62V2

p

(1 ws )41mn +

122V2

p

(1 ws )5n

2m.

To describe the nearness ofn to 3m and to either n or m, we introduce the detuning

parameters 1 and 2 defined by

n = 3m + 21 and = i +

22. (43)


23/27


Because the homogeneous part of Equations (42) and (18) has a nontrivial solution, the non-

homogeneous problem has a solution only if the right-hand side of Equation (42) is orthogonal

to every solution of the adjoint homogeneous problem governing u3 [20]. We note that the

problem is self-adjoint, and hence the adjoints are given by j(x) eijT0 . Multiplying the

right-hand side of Equation (42) by n(x) ein T0 and m(x) e

im T0 , respectively, integrating

the results from x = 0 to x = 1, and using Equations (43), we obtain the following solvability

conditions:

2in(D2An + nAn) = 8SnnA2nAn + 8SnmAnAmAm + 8nA

3m e

i1 T2

+ fn(x)in ei2T2 , (44)

2im(D2Am + mAm) = 8SmmA2mAm + 8SmnAmAnAn + 8mAnA

2m e

i1T2

+ fm(x)im ei2T2 , (45)

where

i = 12

10

c2i dx,

fi =

10

F i dx,

Sii =1

8

1o

1i i dx, Sij =1

8

1o

iji dx, i = j,

n =1

8

1o

5n dx, m =1

8

1o

6m dx.

The Sij are nonlinear coefficients due to electric and geometric sources, and the i are the

nonlinear interaction coefficients between the nth and mth modes.

Expressing the Ai in polar form and separating real and imaginary parts in Equations (44)

and (45), we obtain the modulation equations

an = nan na

3m

nsin 1 +

fnin

nsin 2, (46)

an

n

= Snna

3n

n

Snmana2m

n

na3m

n cos

1

fnin

n cos

2,

(47)

am = mam mana

2m

msin 1 +

fmim

msin 2, (48)

amm =

Smma3m

m

Smnama2n

m

mana2m

mcos 1

fmim

mcos 2, (49)


24/27


where 1 and 2 are defined as

1 = 1T2 3m + n, 2 = 2T2 in n im m.

Consequently, the microbeam dynamic response to second order can be expressed as

u(x,t) = an cos(nt + n)n(x) + am cos(mt + m)m(x)

+1

2a2n [cos 2(nt + n)in(x) + 2n(x)]

+1

2a2m [cos 2(mt + m)im (x) + 2m(x)]

+1

2anam[cos((n + m)t + n + n)3(x)

+ cos((n m)t + n m)4(x)] + , (50)

where an, am, n, and m are governed by Equations (4649).

5.2. RESULTS

We considered the case in which the first mode is directly excited with a primary resonance

(fn = 0) when 2 31. We solved the boundary-value problem, Equations (9), for the static

deflection. Then using this deflection, we solved Equations (13) to determine the first and

second natural frequencies and mode shapes. Using these results, we solved Equations (40)

and (41) for the s and then evaluated the s. Finally, we calculated the Sij, i , and fm.

We considered three cases. The first case corresponds to 1 = 3.70, 2V2

p = 35.30, N = 0,

and the first and the second natural frequencies are 1 = 20.35 and 2 = 61. The second case

corresponds to 1 = 3.70, 2V2

p = 5.50, N = 10, and the natural frequencies are 1 =

19.12 and 2 = 57.70. The last case corresponds to 1 = 3.70, 2V2

p = 56.80, and N = 10,and the natural frequencies are 1 = 21.60 and 2 = 64.20. For all three cases, the numerical

results show that the nonlinear interaction coefficients i vanish identically, which precludesthe possibility of activating this internal resonance. Hence, although the ratio between the

natural frequencies is close to three-to-one, the first and second modes do not exchange energy.

This is because the first mode is symmetric, the second mode is antisymmetric, and the static

deflection is symmetric. And hence, the interaction coefficient m and n are identically zero.

6. Conclusions

We studied the nonlinear dynamic response of a microbeam that is actuated by a general

electric load subject to an applied axial load, accounting for mid-plane stretching. We used

a model that assumes operation at low pressures where the effect of squeeze-film damping

on the resonance frequencies can be neglected. The model does not account for the effectsof shear deformation and rotary inertia. We used the method of multiple scales to determine

the response to a primary resonance excitation of the first mode and obtained two nonlinear

first-order ordinary-differential equations governing the amplitude and phase of the response.

We derived an equation that describes the nonlinear resonance frequency of the microbeam

as a function of the damping and the effective nonlinearity coefficient. This equation shows

that increasing the AC forcing and/or decreasing the damping leads to either an increase


25/27


or a decrease in the nonlinear resonance frequency, depending on the sign of the effective

nonlinearity coefficient. We compared the nonlinear resonance frequency computed using our

theory to the experimental results available in the literature. We found excellent agreement.

We studied the effect of the nondimensional design parameters N, 1, and 2V2

p on the

normalized nonlinear resonance frequency r /. The results show that increasing N, the

axial force, improves the linear characteristics of r / and decreases the frequency shift.

In contrast, increasing 1, the mid-plane stretching, has the reverse effect on r /. On the

other hand, 2V2

p affects the qualitative and quantitative nature of the effective nonlinearity

coefficient of the system. The sign of S changes from positive, corresponding to a hardening

behavior, to negative corresponding to a softening behavior. This is because the electric non-

linearity, mainly the quadratic, drastically increases in magnitude and overcomes the influence

of the geometric nonlinearity. We note that most of the models used in the literature neglect

the effect of the electric nonlinearity and particularly the quadratic one. Instead, they assume

the nonlinearity of the system to be solely cubic and positive, which predicts a harding beha-

vior rather than the correct softening behavior. Therefore, failure to correctly account for the

nonlinearities in the system may lead to erroneous results. It is interesting to note that, for the

studied case, this reverse in behavior occurs at a value of 2V2

p , which is about 46% of the

pull-in limit. Because at this value the coefficient S is equal to zero, the dynamic response istheoretically linear. However, the simulation results show that r / becomes highly sensitive

to any slight changes in 2V2

p . This indicates that, although it is possible to compensate for the

effect of the electrostatic force with mid-plane stretching, operations near such conditions are

unstable and impractical. We believe more experimental work is needed to better understand

the system behavior near this region.

We applied the method of multiple scales to investigate possibility of activating a three-to-

one internal resonance between the first and second modes, which if exists can adversely affect

the performance of the resonator. The analysis shows that these two modes are nonlinearly

uncoupled, and hence this internal resonance cannot be activated. This result enhances the

reliability of such a device.

In conclusion, the present nonlinear model provides an accurate prediction of the dynamic

behavior of microbeams, which linear models fail to explain. Unlike existing models in the

literature, the present nonlinear model is capable of simulating the mechanical behavior of

microbeams for general operating conditions and for a wider range of applied electric loads.

Further, by using the perturbation solution, one can easily derive analytical expressions that

present a clearer picture of the influence of various design parameters, which is critical for

improving and optimizing designs.

Appendix A: Note on Energy Methods

We consider the microbeam shown in Figure 1 with zero AC and DC forces. To obtain the

equation that governs the microbeam motion, we set Vp and v(t) equal to zero in Equation (3),

assume the same boundary conditions as in Equation (4), and obtainEI

1 24w

x4+ bh

2w

t2+ c

w

t

=

EA

2(1 2)

0

w

x

2dx + N

2w

x2, (A.1a)


26/27


w(0, t) = w(,t) = 0 andw

x

(0,t )

=w

x

(,t)

= 0. (A.1b)

Equations (A.1) can be derived using the EulerLagrange equations from the Lagrangian

L = T U, (A.2)

where the potential U and kinetic T energies of the system are given by

U =EI

2(1 2)

0

2w

x2

2dx +

1

2N

0

w

x

2dx

+EA

8(1 2)

0

w

x

2dx

2

, (A.3a)

T =

1

2 bh

0

wt

2dx. (A.3b)

The total energy of the system is given by

H = U + T . (A.4)

Studying the same problem, Tilmans et al. [14] misrepresented the potential energy due to

mid-plane stretching by expressing it as

Umid =EA

8

0

w

x

4dx.

The correct representation however is the last term in Equation (A.3a). The theories of Tilmans

and Legtenburg [9] and Gui et al. [10] are based on this incorrect representation of the potential

energy.

References

1. Goldsmith, C. L., Yao, Z., Eshelman, S., and Denniston, D., Performance of low-loss RF MEMS capacitive

switches, IEEE Microwave and Guided Wave Letters 8, 1998, 269271.

2. Chan, E. K., Garikipati, K., and Dutton, W. R., Characteristics of contact electromechancis through

capacitance-voltage measurements and simulations, Journal of Microelectromechanical Systems 8, 1999,

208217.

3. Gupta, R. K. and Senturia, S. D., Pull-in time dynamics as a measure of absolute pressure, in Proceedingsof the IEEE Tenth Annual International Workshop on Microelectromechanical Systems: MEMS 97, Nagoya,

Japan, IEEE, New York, 1997, pp. 5159.

4. Andrews, M. K. , Turner, G. C., Harris, P. D., and Harris, I. M., A resonant pressure sensor based on a

squeezed film of gas, Sensors and Actuators A36, 1993, 219226.

5. Bouwstra, S. and Geijselaers, B., On the resonance frequencies of microbridges, in Proceedings of the 6th

International Conference on Solid-State Sensors and Actuators (TRANSDUCERS 91), San Francisco, CA,

IEEE, New York, 1997, Vol. 2, pp. 11411144.


27/27


6. Younis, M. I., Abdel-Rahman, E. M., and Nayfeh, A. H., A reduced-order model for electrically actuated

microbeam-based MEMS, Journal of Microelectromechanical Systems, to appear.

7. Younis, M. I., Abdel-Rahman, E. M., and Nayfeh, A. H., Static and dynamic behavior of an electrically ex-

cited resonant microbeam, in Proceedings of the AIAA 43rd Structures, Structural Dynamics, and Materials

Conference, Denver, CO, 2002, AIAA Paper No. 2002-1305.

8. Zook, J. D., Burns, D. W., Guckel, H., Sniegowski, J. J., Engelstad, R. L., and Feng, Z., Characteristics of

polysilicon resonant microbeams, Sensors and Actuators A35, 1992, 290294.

9. Tilmans, H. A. and Legtenberg, R., Electrostatically driven vacuum-encapsulated polysilicon resonators.

Part II. Theory and performance, Sensors and Actuators A45, 1994, 6784.

10. Gui, C., Legtenberg, R., Tilmans, H. A., Fluitman, J. H., and Elwenspoek, M.,Nonlinearity and hysteresis

of resonant strain gauges, Journal of Microelectromechanical Systems 7, 1998, 122127.

11. Turner, G. C. and Andrews, M. K., Frequency stabilization of electrostatic oscillators, in Digest of the 8th

International Conference on Solid-State Sensors and Actuators, Vol. 2, Stockholm, Sweden, S. Middelhoek

and K. Cammann (eds.), Elsevier, Amsterdam, 1995, Vol. 2, pp. 624626.

12. Ayela, F. and Fournier, T., An experimental study of anharmonic micromachined silicon resonators,

Measurement, Science and Technology 9, 1998, 18211830.

13. Veijola, T., Mattila, T., Jaakkola, O., Kiihamki, J., Lamminmki, T., Oja, A., Ruokonen, K., Sep, H.,

Seppl, P., and Tittonen, I., Large-displacement modeling and simulation of micromechanical electrostat-

ically driven resonators using the harmonic balance method, in the IEEE MTT-S International Microwave

Symposium Digest, Boston, MA, T. Perkins (ed.), IEEE, New York, 2000, Vol. 1, pp. 99102.

14. Tilmans, H. A., Elwespoek, M., and Fluitman, J. H., Micro resonant force gauges, Sensors and ActuatorsA30, 1992, 3553.

15. Gui, C., Legtenberg, R., Elwenspoek, M., and Fluitman, J. H., Q-factor dependence of one-port encapsulated

polysilicon resonator on reactive sealing pressure, Journal of Micromechanics and Microengineering 5,

1995, 183185.

16. Legtenberg, R. and Tilmans, H. A., Electrostatically driven vacuum-encapsulated polysilicon resonators.

Part I. Design and fabrication, Sensors and Actuators A45, 1994, 5766.

17. Nayfeh, A. H., Nonlinear Interactions, Wiley, New York, 2000.

18. Griffiths., D. J., Introduction to Electrodynamics, Prentice Hall, Englewood Cliffs, NJ, 1981.

19. Meng, Q., Mehregany, M., and Mullen, R., Theoretical modeling of microfabricated beams with elastically

restrained supports, Journal of Microelectromechanical Systems 2, 1993, 128137.

20. Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley, New York, 1981.

21. Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.

22. Nayfeh, A. H. and Balachandran B., Applied Nonlinear Dynamics, Wiley, New York, 1995.

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Resonant micro beam