The Effect of Residual Stress on the Electromechanical ...downloads.hindawi.com/journals/apec/2008/905628.pdf · A shaped microbeam with a curved electrode is shown in Figure 1. The

Hindawi Publishing CorporationActive and Passive Electronic ComponentsVolume 2008, Article ID 905628, 9 pagesdoi:10.1155/2008/905628

Research ArticleThe Effect of Residual Stress on the ElectromechanicalBehavior of Electrostatic Microactuators

Ming-Hung Hsu

Department of Electrical Engineering, National Penghu University, Penghu 880, Taiwan

Correspondence should be addressed to Ming-Hung Hsu, [email protected]

Received 22 October 2008; Accepted 21 December 2008

Recommended by Yalin Lu

This work simulates the nonlinear electromechanical behavior of different electrostatic microactuators. It applies the differentialquadrature method, Hamilton’s principle, and Wilson-θ integration method to derive the equations of motion of electrostaticmicroactuators and find a solution to these equations. Nonlinear equation difficulties are overcome by using the differentialquadrature method. The stresses of electrostatic actuators are determined, and the residual stress effects of electrostaticmicroactuators are simulated.

Copyright © 2008 Ming-Hung Hsu. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

Osterberg et al. [1] analyzed electrostatically deformeddiaphragms using a one-dimensional numerical model anda three-dimensional model. Osterberg and Senturia [2]showed that the sharp instability phenomena of electrostaticpull-in behavior of cantilever beam and fixed-fixed beamactuators can be adopted to extract the material propertiesof microelectromechanical system. Elwenspoek et al. [3]studied the dynamic behavior of active joints for variouselectrostatic actuator designs. Hirai et al. [4–6] presentedthe deflection characteristics of electrostatic actuators withmodified electrode and cantilever shapes. Wang [7] applieda feedback control for suppressing the vibration of actuatorbeams in an electrostatic actuator. Shi et al. [8] combinedan exterior boundary element method for electrostaticsand a finite element method for elasticity to evaluate thecoupling effect between the electrostatic force and the elasticdeformation. Gretillat et al. [9] employed three-dimensionalfinite element programs to simulate the dynamics of anonlinear actuator, considering the effect of squeeze-filmdamping. Hung and Senturia [10] proposed leveragedbending and strain-stiffening methods to enlarge the limit oftravel distance before pull-in of electrostatic actuators. Thiswork will analyze the nonlinear pull-in behaviors of differenttypes of microactuator with various residual stresses using

the differential quadrature method. The Chebyshev-Gauss-Lobatto point distribution on each actuator will be used.The integrity and computational accuracy of the differentialquadrature method in solving this problem will be evaluatedthrough a range of case studies. The dynamic equationsof the cantilever microactuator are derived using the dif-ferential quadrature method. The equations describing theresidual vibrations of the microelectrostatic actuators arederived in this paper. The differential quadrature method isused to produce the electrostatic field equations in matrixform.

2. THE DIFFERENTIAL QUADRATURE METHOD

This paper employs the differential quadrature method,with its easy-to-use and meshless technique, to analyzethe nonlinear deflection behaviors of different types ofmicroactuator with different residual stresses. There area number of solution techniques for complicated beamproblems, such as the Rayleigh-Ritz method, the analyticalmethod, the Galerkin method, the finite element method,and the boundary element method. The differential quadra-ture method has been extensively used to solve a variety ofproblems in different fields of science and engineering withno need of energy formulation. The differential quadraturemethod has been shown to be a powerful contender in

2 Active and Passive Electronic Components

solving initial and boundary value problems and has, thus,become an alternative to the previous methods. Jang et al.[11] proposed the δ method, in which the boundary pointsare selected at a small distance from each other. The δtechnique can be applied to the double boundary conditionsof plate and beam problems. The accuracy of the solutiondepends on a sufficiently small δ. The boundary pointsare chosen at a small distance δ. The δ technique can beapplied to the double boundary conditions of plate and beamproblems. The use of δ at the boundary makes the matrixill conditioned [11]. Wang and Bert [12] considered theboundary conditions in finding the differential quadratureweighting coefficients. Malik and Bert [13] solved theproblem of the free vibration of the plates and showed thatthe boundary conditions can be built into the differentialquadrature weighting coefficients. In their formulation, themultiple boundary conditions are directly applied to thedifferential quadrature weighting coefficients, and thus itis not necessary to select a nearby point. In other words,the accuracy of the calculated results will be independentof the value of the δ-interval. The differential quadratureweighting coefficients can be obtained by multiplying theinverse matrix [13]. Sherbourne and Pandey [14] solvedbuckling problems using the differential quadrature method.From the foregoing discussion, over the last two decades, thedifferential quadrature method has been applied extensivelyas an effective means of solving a range of problems invarious fields of science and engineering. Quan and Chang[15, 16] derived the weighting coefficients in a more explicitway. Feng and Bert [17] analyzed the flexural vibrationanalysis of a geometrically nonlinear beam using the quadra-ture method. Chen and Zhong [18] presented the studyon the nonlinear computations of the differential quadra-ture method and differential cubature method. Tomasiello[19] applied the differential quadrature method to initial-boundary-value problems. Wang et al. [20] presented the freevibration analysis of circular annular plates with nonuniformthickness by the differential quadrature method. Wang andGu [21] presented the static analysis of frame structuresby the differential quadrature element method. Liew et al.[22, 23] presented the differential quadrature method forMindlin plates on Winkler foundations and thick symmetriccross-ply laminates with first-order shear flexibility. Du et al.[24] presented the application of a generalized differentialquadrature method to structural problems. Mirfakhraei andRedekop [25] solved the buckling of circular cylindrical shellsusing the differential quadrature method. Moradi and Taheri[26] presented the delamination buckling analysis of generallaminated composite beams using the differential quadraturemethod. De Rosa and Franciosi [27] introduced the exactand approximate dynamic analysis of circular arches usingthe differential quadrature method. Sun and Zhu [28]used the upwind local differential quadrature method forsolving incompressible viscous flow. Gu and Wang [29]presented the free vibration analysis of circular plates withstepped thickness over a concentric region by the differentialquadrature method. Du et al. [30] presented the generalizeddifferential quadrature method for buckling analysis. Hanand Liew [31] analyzed axisymmetric free vibration of thick

annular plates. Tanaka and Chen [32] applied a dual reci-procity boundaryelement method to transient elastodynamicproblems using the differential quadrature method. Chenet al. [33] solved the high-accuracy plane stress and plateelements by the quadrature element method. The essenceof the differential quadrature method is that the derivativeof a function at a sample point can be approximatedas a weighted linear summation of the functional valuesat all of the sampling points in the domain. Using thisapproximation, the differential equation is then reduced to aset of algebraic equations. The effects of position-dependentelectrostatic force and axial residual stress have all beenconsidered in the proposed models. While the efficiency andaccuracy of the Rayleigh-Ritz method depend on the numberand accuracy of the selected comparison functions; thedifferential quadrature method does not have this difficultyof selecting the appropriate comparison functions. Thedifferential quadrature method approximates the mth orderpartial derivative of f (z, t) with respect to z. For a functionf (z, t), the differential quadrature approximation for themth order derivative at the ith sampling point is given by

∂m

∂zm

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

f (z1, t)

f (z2, t)...

f (zN , t)

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

∼=[D(m)i j

]

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

f (z1, t)

f (z2, t)...

f (zN , t)

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

for i, j = 1, 2, . . . N ,

(1)

in which f (zi, t) is the functional value at the sample

point zi, and D(m)i j are the differential quadrature weighting

coefficients of the mth order differentiation attached to thesefunctional values. Quan and Chang [15, 16] introduceda Lagrangian interpolation polynomial to overcome thenumerical ill conditioning in determining the differential

quadrature weighting coefficients D(m)i j , that is,

f (z, t) =N∑

i=1

M(z)(z − zi)M1(zi)

f (zi, t), (2)

where

M(z) =N∏

j=1

(z − zj),

M1(zi) =N∏

j=1, j /= i(zi − zj) for i = 1, 2, . . . ,N.

(3)

Equation (2) is substituted into (1). The differential quadra-ture weighting coefficients are then given as

D(1)i j =

M1(zi)(zi − zj)M1(zj)

for i, j = 1, 2, . . . ,N , i /= j,

D(1)ii = −

N∑

j=1, j /= iD(1)i j for i = 1, 2, . . . ,N.

(4)

Once the sampling points, such as zi for i = 1, 2, . . . ,N ,are selected, the coefficients of the differential quadrature

Ming-Hung Hsu 3

weighting matrix can be obtained from (4). Higher orderderivatives of the differential quadrature weighting coeffi-cients can also be directly calculated by matrix multiplication[34], which can be expressed as

D(2)i j =

N∑

k=1

D(1)ik D

(1)k j for i, j = 1, 2, . . . ,N ,

D(3)i j =

N∑

k=1

D(1)ik D

(2)k j for i, j = 1, 2, . . . ,N ,

D(4)i j =

N∑

k=1

D(1)ik D

(3)k j for i, j = 1, 2, . . . ,N ,

...

D(m)i j =

N∑

k=1

D(1)ik D

(m−1)k j for i, j = 1, 2, . . . ,N.

(5)

There are many computational methods available fordynamic analysis. In this paper, the residual vibrations of themicroelectrostatic actuators are investigated using the differ-ential quadrature method. The most convenient approachto solving a beam structure problem is to uniformly spaceout the sample points. The selection of sample points isimportant for the accuracy of the differential quadraturemethod solution, but inaccurate results have been obtainedwhen using this uniform distribution. A nonuniform samplepoint distribution, such as the Chebyshev-Gauss-Lobattodistribution [34], improves the accuracy of the calculation.The integrity and computational efficiency of the differentialquadrature method in solving this problem will be demon-strated using a set of case studies. However, an alternativeefficient technique is still sought. In this study, the unequallyspaced sample points of each beam using the Chebyshev-Gauss-Lobatto distribution are chosen as

zi = L

2

(

1− cos(i− 1)πN − 1

)

for i = 1, 2, . . . ,N. (6)

The differential quadrature method has been shown to be apowerful candidate for solving initial and boundary valueproblems, and has thus become an alternative to othermethods. The efficiency and the accuracy of Rayleigh-Ritzmethod depend on the number and accuracy of the selectedcomparison functions, whereas the differential quadraturemethod does not have such a difficulty. Like that of anypolynomial approach, the accuracy of the solution using thismethod is improved by increasing the number of samplepoints. The differential quadrature method uses high-orderelement level, where the finite element method approximatesa function using low-order polynomials.

3. DYNAMIC BEHAVIOR OF MICROACTUATORS

A shaped microbeam with a curved electrode is shownin Figure 1. The figure depicts the geometry of a taperedelectrostatic microactuator. t0 specifies the thickness at theroot of the actuator. L is the length of the microactuator. q(z)is the load. Load q(z) acts on z = 0∼L in the beam. As adriving voltage is applied between the fixed-fixed microbeam

Electrode

y

+

V

−

d

δe

t0

q(z)

Actuator

Dielectricstructure

z

L

Figure 1: Schematic view of an electrostatic fixed-fixed actuator.

and the electrode, a position-dependent electrostatic pres-sure is distributed to deform the microbeam toward thecurved electrode. The gap between the shaped microbeamand the curved electrode determines the distribution of theelectrostatic pressure. To prevent a short circuit after pull-in contact, an isolated layer or other structure is required.The force pulls the microbeam toward the shaped electrode.Different electrode shapes have been proposed to improvethe electrostatic force distribution and the deformed shapeof the actuator. The kinetic energy of the microactuator is

T = 12

∫ L

0ρA(∂u

∂t

)2

dz +12

∫ L

0ρA(∂v

∂t

)2

dz

+12

∫ L

0GJz

(∂φ

∂t

)2

dz,

(7)

where u is the displacement in the direction of the x-axis,v is the displacement in the direction of the y-axis, φ is thetwist angle in the direction of the z-axis, A is the area of thecross-section of the microbeam, Jz is the polar moment in thedirection of the z-axis, and ρ is the density of the material ofthe actuator.

While the external voltage e is applied between thedeformable beam and the fixed electrode, a position-dependent electrostatic pressure is created to pull thedeformable beam toward the ground electrode. This electro-static pressure is approximately proportional to the inverseof the square of the gap between them. When the voltagereaches the critical voltage, the fixed-fixed beam will bepulled toward the electrode suddenly. The electric fringingeffects are ignored in the following analyses. The strainenergy of the microactuator can be approximated as

U = 12

∫ L

0E(

Iyy

(∂2u

∂z2

)2

+ 2Ixy

(∂u

∂z

)(∂v

∂z

)

+ Ixx

(∂2v

∂z2

)2)

dz

+12

∫ L

0GJz

(∂φ

∂z

)2

dz

− 12

∫ L

0P(∂u

∂z

)2

dz − 12

∫ L

0P(∂v

∂z

)2

dz,

(8)


where E is Young’s modulus of the actuator, G is the shearmodulus, and Ixx, Iyy , and Ixy are the moments of area. Theload P is the residual axial loading acting on the fixed end ofthe actuator. The value of P is σb0t0. σ is the residual stress,and b0 is the beam width. Because of the coupling betweenthe mechanical and electrostatic effects, the behavior ofthe electrostatic actuator appears more complicated thanelastic behavior. The external damping presents a viscousresistance to transverse displacement of the actuator, and theinternal damping provides a viscous resistance to straining ofthe microactuator material. The damping forces cu(∂u/∂t),cv(∂v/∂t), and cφ(∂φ/∂t) are assumed for resistance to thetransverse velocity of the actuator. The damping forcescui(∂2/∂z2)(EI(∂3u/∂t∂z2)), cvi(∂2/∂z2)(EI(∂3v/∂t∂z2)), andcφi(∂/∂z)(GJz(∂2φ/∂t∂z)) are assumed for the resistance tothe strain velocity of the microactuator. Considering theelectrostatic force and the internal and external dampingeffects in the actuator, the virtual work δW done by the bentactuator is

δW = −∫ L

0cu∂u

∂tδu dz −

∫ L

0cui

∂2

∂z2

(

EI∂3u

∂t∂z2

)

δudz

−∫ L

0cv∂v

∂tδv dz −

∫ L

0cvi

∂2

∂z2

(

EI∂3v

∂t∂z2

)

δv dz

−∫ L

0cφ∂φ

∂tδφ dz +

∫ L

0cφi

∂

∂z

(

GJz∂2φ

∂t∂z

)

δφ dz

+∫ L

0

ε0b0e2

2(d + S− α sin(πz/L)t0/2− v)2 δv dz

−∫ L

0q(z)δv dz,

(9)

where e is the applied voltage, ε0 is the dielectric constantof air, such as ε0 = 8.85 × 10−12, b0 is the width of theactuator, and d is the initial gap as shown in Figure 1.The cross-section area of the actuator is A(z) = b0t0(1 +α sin(πz/L)), α is the constant. I(z) is the moment of inertiaof the cross-sectional area of the actuator, which is I(z) =I0(1 + α sin(πz/L))3 and I0 = b0t

30/12. The shape function

S(z) describes the shape of the curved electrode, and it ispresented as S(z) = δe + β sin(πz/L). δe is the fixed endgap distance of the curved electrode at z = 0 and z = L.The electrode shape is varied with the values of β and δe.However, due to the difficulty of no linearity between theactuator deflection and the electrostatic force, this residualvibration phenomenon has been studied in only a very fewpapers, as has the effect of electrode shape on the residualresponse. Substituting (7), (8), and (9) into Hamilton’sequation:

∫ t2

t1(δT − δU + δW)dt = 0, (10)

the dynamic deflection of a fixed-fixed micro-actuator can beexpressed as the following nonlinear differential equation:

E∂2Iyy∂z2

∂2u

∂z2+ 2E

∂Iyy∂z

∂3u

∂z3+ EIyy

∂4u

∂z4+ E

∂2Ixy∂z2

∂2v

∂z2

+ 2E∂Ixy∂z

∂3v

∂z3+ EIxy

∂4v

∂z4+ P

∂2v

∂z2

+ cu∂u

∂t+ cui

∂2

∂z2

(

EIyy∂3u

∂z2∂t

)

+ ρA∂2u

∂t2= 0,

E∂2Ixx∂z2

∂2v

∂z2+ 2E

∂Ixx∂z

∂3v

∂z3+ EIxx

∂4v

∂z4+ E

∂2Ixy∂z2

∂2u

∂z2

+ 2E∂Ixy∂z

∂3u

∂z3+ EIxy

∂4u

∂z4+ P

∂2v

∂z2

+ cv∂v

∂t+ cvi

∂2

∂z2

(

EIxx∂3v

∂z2∂t

)

+ ρA∂2v

∂t2

= ε0b0e2

2(d + S(z)− α sin(πz/L)t0/2− v(z))2 − q(z)

− ∂

∂z

(

GJz∂φ

∂z

)

+ cφ∂φ

∂t− cφi ∂

∂z

(

GJz∂2φ

∂t∂z

)

+ ρJz∂2φ

∂t2= 0,

(11)

where ε0 is the dielectric constant of air. The correspondingboundary conditions of the clamped-clamped micro-ctuatorare

u(0, t) = 0,

∂u(0, t)∂z

= 0,

u(L, t) = 0,

∂u(L, t)∂z

= 0,

v(0, t) = 0,

∂v(0, t)∂z

= 0,

v(L, t) = 0,

∂v(L, t)∂z

= 0,

φ(0, t) = 0,

φ(L, t) = 0.

(12)

Equation (1) is substituted into (11)-(12) by employing thedifferential quadrature method. The equations of motion ofthe microactuator can be discretized in matrix form withrespect to the sample points as

[M]{∂2w

∂t2

}

+ [C]{∂w

∂t

}

+ [K]{w} = {F}. (13)

Ming-Hung Hsu 5

The displacement vector at the sample points is

{w} =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

u(z1)

u(z2)

...

u(zN )

v(z1)

v(z2)

...

v(zN )

φ(z1)

φ(z2)

...

φ(zN )

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (14)

The elements in the mass matrix are

Mii = 0 for i = 1, 2,

Mii = ρA for i = 3, 4, . . . ,N − 2,

Mii = 0 for i = N − 1,N ,

Mij = 0 for i /= j, i = 1, 2, . . . ,N , j = 1, 2, . . . ,N ,

Mij = 0 for i = 1, 2, . . . ,N , j = N + 1,N + 2, . . . , 2N ,

Mij = 0 for i = 1, 2, . . . ,N , j = 2N + 1, 2N + 2, . . . , 3N ,

Mij = 0 for i = N + 1,N + 2, . . . , 2N , j = 1, 2, . . . ,N ,

Mii = 0 for i = N + 1,N + 2,

Mii = ρA for i = N + 3,N + 4, . . . , 2N − 2,

Mii = 0 for i = 2N − 1, 2N ,

Mij = 0 for i /= j, i = N + 1,N + 2, . . . , 2N ,

j = N + 1,N + 2, . . . , 2N ,

Mij = 0 for i = N + 1,N + 2, . . . , 2N ,

j = 2N + 1, 2N + 2, . . . , 3N ,

Mij = 0 for i = 2N + 1, 2N + 2, . . . , 3N ,

j = 1, 2, . . . ,N ,

Mij = 0 for i = 2N + 1, 2N + 2, . . . , 3N ,

j = N + 1,N + 2, . . . , 2N ,

Mii = 0 for i = 2N + 1,

Mii = ρJz for i = 2N + 2, 2N + 3, . . . , 3N − 1,

Mii = 0 for i = 3N ,

Mij = 0 for i /= j, i = 2N + 1, 2N + 2, . . . , 3N ,

j = 2N + 1, 2N + 2, . . . , 3N.(15)

The elements in the damping matrix are

Cij = 0 for i = 1, 2, j = 1, 2, . . . , 3N ,

Cii = cu + cui∂2

∂z2(EIyy)D

(2)ii + 2cui

∂

∂z(EIyy)D

(3)ii

+ cuiEIyyD(4)ii for i = 3, 4, . . . ,N − 2,

Cij = cui∂2

∂z2(EIyy)D

(2)i j + 2cui

∂

∂z(EIyy)D

(3)i j + cuiEIyyD

(4)i j

for i /= j, i = 3, 4, . . . ,N − 2, j = 1, 2, . . . ,N ,

Cij = 0 for i = 3, 4, . . . ,N − 3,N − 2,

j = N + 1,N + 2, . . . , 3N ,

Cij = 0 for i = N − 1,N , j = 1, 2, . . . , 3N ,

Cij = 0 for i = N + 1,N + 2, . . . , 2N , j = 1, 2, . . . ,N ,

Cij = 0 for i = N + 1,N + 2, j = N + 1,N + 2, . . . , 2N ,

Cii = cv + cvi∂2

∂z2(EIxx)D(2)

ii + 2cvi∂

∂z(EIxx)D(3)

ii + cviEIxxD(4)ii

for i = N + 3,N + 4, . . . , 2N − 2,

Cij = cvi∂2

∂z2(EIxx)D(2)

i j + 2cvi∂

∂z(EIxx)D(3)

i j + cviEIxxD(4)i j

for i /= j, i = N+3,N+4, . . . , 2N − 2,

j = N + 1,N+2, . . . , 2N ,

Cij = 0 for i = 2N − 1, 2N , j = N + 1,N + 2, . . . , 2N ,

Cij = 0 for i = N + 1,N + 2, . . . , 2N ,

j = 2N + 1, 2N + 2, . . . , 3N ,

Cij = 0 for i = 2N + 1, j = 1, 2, . . . , 3N ,

Cij = 0 for i = 2N + 2, 2N + 3, . . . , 3N − 1,

j = 1, 2, . . . , 2N ,

Cii = cφ − cφi ∂∂z

(GJz)D(1)ii − cφi(GJz)D(2)

ii

for i = 2N + 2, 2N + 3, . . . , 3N − 1,

Cij = −cφi ∂∂z

(GJz)D(1)i j − cφi(GJz)D(2)

i j

for i /= j, i = 2N + 2, 2N + 3, . . . , 3N − 1,

j = 2N + 1, 2N + 2, . . . , 3N ,

Cij = 0 for i = 3N , j = 1, 2, . . . , 3N.(16)


The elements in the stiffness matrices are

K11 = 1,

K1 j = 0 for j = 2, 3, . . . , 3N ,

K2 j =D(1)

1 j

Lfor j = 1, 2, . . . ,N ,

K2 j = 0 for j = N + 1,N + 2, . . . , 3N ,

Kij = E∂2Iyy∂z2

∣∣∣∣z=zi

D(2)i j + 2E

∂Iyy∂z

∣∣∣∣z=zi

D(3)i j

+ EIyyD(4)i j + PD(2)

i j for i = 3, 4, . . . ,N − 2,

j = 1, 2, . . . ,N ,

Kij = E∂2Ixy∂z2

∣∣∣∣z=zi

D(2)i, j−N + 2E

∂Ixy∂z

∣∣∣∣z=zi

D(3)i, j−N

+ EIxyD(4)i, j−N for i = 3, 4, . . . ,N − 2,

j = N + 1,N + 2, . . . , 2N ,

KN−1, j = D(1)N , j−N for j = 1, 2, . . . ,N ,

KN j = 0 for j = 1, 2, . . . ,N − 1,

KNN = 1,

KN j = 0 for j = N + 1,N + 2, . . . , 3N ,

Kij = 0 for i = 3, 4, . . . ,N − 2,

j = 2N + 1, 2N + 2, . . . , 3N ,

Kij = 0 for i = N + 1, j = 1, 2, . . . ,N ,

KN+1,N+1 = 1,

Kij = 0 for i = N + 1, i = N + 2,N + 3, . . . , 3N ,

Kij = 0 for i = N + 2, j = 1, 2, . . . ,N ,

Kij = D(1)1, j−N for i = N + 2, j = N + 1,N + 2, . . . , 2N ,

Kij = 0 for i = N + 2, i = 2N + 1, 2N + 2, . . . , 3N ,

Kij = E∂2Ixy∂z2

∣∣∣∣z=zi

D(2)i−N , j + 2E

∂Ixy∂z

∣∣∣∣z=zi

D(3)i−N , j + EIyyD

(4)ii

for i = N + 3,N + 4, . . . , 2N − 2, j = 1, 2, . . . ,N ,

Kij = E∂2Ixx∂z2

∣∣∣∣z=zi

D(2)i−N , j−N + 2E

∂Ixx∂z

∣∣∣∣z=zi

D(3)i−N , j−N

+ IxxD(4)i−N , j−N for i = N + 3,N + 4, . . . , 2N − 2,

i = N + 1,N + 2, . . . , 2N ,

Kij = 0 for i = 2N − 1, j = 1, 2, . . . ,N ,

Kij = D(1)1, j−N for i = 2N − 1, j = N + 1,N + 2, . . . , 2N ,

Kij = 0 for i = 2N − 1, j = 2N + 1, 2N + 2, . . . , 3N ,

Kij = 0 for i = 2N + 1, j = 1, 2, . . . , 2N ,

Kij = 1 for i = 2N + 1, j = 2N + 1,

Kij = 0 for i = 2N + 1, j = 2N + 2, . . . , 3N ,

Kij = 0 for i = 2N + 2, 2N + 3, . . . , 3N ,

j = 1, 2, . . . , 2N ,

Kij = −G∂Jz∂z

∣∣∣∣z=zi

D(1)i−2N , j−2N −GJzD(2)

i−2N , j−2N

for i = 2N + 2, 2N + 3, . . . , 3N − 1,

j = 2N + 1, 2N + 2, . . . , 3N ,

Kij = 0 for i = 3N , j = 1, 2, . . . , 3N − 1,

Kij = 1 for i = 3N , j = 3N ,

Fi = 0 for i = 1, 2, . . . ,N + 2,

Fi = ε0be2

2(d + S(z)− α sin(πz/L)t0/2− v(z))2 − q(z)

for i = N + 3,N + 4, . . . , 2N − 2,

Fi = 0 for i = 2N − 1, 2N , . . . , 3N.

(17)

The dynamic responses of the microactuator are solvedusing the Wilson-θ integration method in this paper. TheWilson-θ integration method is an effective implicit timeintegration procedure for dynamic problems. It is a step-by-step integration method that assumes that the accelerationterms vary linearly between consecutive sampling instants.An electrostatic force pulls the cantilever actuator toward thecurved electrode. The electrostatic force is generated by thedifference between voltage applied to the curved electrodeand that applied to the actuator. This electrostatic pressureis approximately proportional to the inverse of the square ofthe gap between them. When the voltage exceeds the criticalvoltage, the fixed-fixed beam is suddenly pulled into theelectrode.

4. NUMERICAL RESULTS AND DISCUSSION

The microactuator is fabricated from polysilicon material.The geometric parameters and the material of the microac-tuator are E = 150 GPa, δmax = 30 μm, α = 0, β = 0, cui = 0,cvi = 0, cφi = 0, cu = 0, cv = 0, cφ = 0, b0 = 5 μm,t0 = 2 μm, d = 2 μm, and L = 500 μm. Figure 2 showsthe deflections of the microbeam with different appliedvoltages. The results indicate that the results calculated fromthe proposed differential quadrature method agree very wellwith the results found using the finite element method.Figure 3 shows the frequencies of an electrostatic fixed-fixedactuator for various lengths of the beam. Again, the resultsfound using the differential quadrature method are similarto the results found using the finite element method. Figure 4plots the deflections near the middle of an electrostatic fixed-fixed actuator for various residual stresses. The value ofapplied voltageis 620 V. The nonlinear dynamic equationformed by the differential quadrature method is solved by

Ming-Hung Hsu 7

6005004003002001000

Length (μm)

620 V (present)310 V (present)

620 V (finite element method)310 V (finite element method)

0

0.5

1

1.5

2

2.5

3

3.5

Defl

ecti

on(μ

m)

Figure 2: Deflections of an electrostatic fixed-fixed actuator forvarious applied voltages.

543210

Mode number

Present (L = 500μm)Finite element method (L = 500μm)Present (L = 400μm)Finite element method (L = 400μm)Present (L = 300μm)Finite element method (L = 300μm)

0

1

2

3

4

5

6

7

8

9

10×105

Freq

uen

cy(H

z)

Figure 3: Frequencies of an electrostatic fixed-fixed actuator forvarious beam lengths.

543210

Time (ms)

0 N/m/m10 N/m/m

20 N/m/m30 N/m/m

0

1

2

3

4

5

6

Defl

ecti

on(μ

m)

Figure 4: Deflections near the middle of an electrostatic fixed-fixedactuator for various residual stresses.

109876543210

Time (ms)

0 N/m/m10 N/m/m

20 N/m/m30 N/m/m

0

5

10

15

20

25

30

35

40

Stre

ss(N

/m/m

)

Figure 5: Stresses near the middle of an electrostatic fixed-fixedactuator for various residual stresses.

1086420

Time (ms)

0 N/m/m10 N/m/m

20 N/m/m30 N/m/m

0

5

10

15

20

25

30

35

40

45

Stre

ss(N

/m/m

)

Figure 6: Stresses near the root of an electrostatic fixed-fixedactuator for various residual stresses.

the Wilson-θ integration method, with θ = 1.4 and Δt =0.003 millisecond. A number of papers state that Wilson-θintegration method is unconditionally stable with a factorof θ ≥ 1.37 [35, 36]. The calculated results show thathigher residual stresses produce smaller deflections near themiddle of an electrostatic fixed-fixed actuator. Figure 5 showsthe stresses near the middle of an electrostatic fixed-fixedactuator for various residual stresses. Numerical results inthis example show that the residual stresses can significantlyaffect the dynamic behavior of the actuator system, showingthat higher residual stresses produce larger stresses near themiddle of an electrostatic fixed-fixed actuator. Figure 6 shows


the stress near the root of an electrostatic fixed-fixed actuatorfor various residual stresses. Results indicate that residualstress is a very sensitive parameter for the residual vibrationof the microactuator. Numerical results in this example showthat the driving voltage can affect the electromechanicalbehavior of the actuator system significantly. Calculatedresults also display that the higher residual stresses introducethe larger stresses near the root of an electrostatic fixed-fixed actuator. Residual axial loading should be consideredin the design. Numerical results indicate that the differentialquadrature method is a feasible and efficient method toanalyze the nonlinear pull-in behavior of a fixed-fixed typeof electrostatic microbeam.

5. CONCLUSIONS

The differential quadrature method is highly suited todesigning or analyzing an electrostatic microactuator. Thesimplicity of this formulation makes it a strong candidate formodeling applications that are more complicated. The effectsof residual stresses of microactuators on the nonlinear pull-in phenomena have also been investigated by employing theproposed differential quadrature method algorithm.

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The Effect of Residual Stress on the Electromechanical ...downloads.hindawi.com/journals/apec/2008/905628.pdf · A shaped microbeam with a curved electrode is shown in Figure 1. The