# The Effect of Residual Stress on the Electromechanical Behavior of Electrostatic Microactuators

Authors: Ming-Hung Hsu
Journal: Active and Passive Electronic Components
 (2008)

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## Abstract

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


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## Body

## 1. Introduction

Osterberg et al. [1] analyzed electrostatically deformed diaphragms using a
one-dimensional
numerical
model and a three-dimensional model. Osterberg and
Senturia [2] showed that the sharp instability phenomena of electrostatic
pull-in behavior of cantilever beam and fixed-fixed beam actuators can be
adopted to extract the material properties of microelectromechanical system.
Elwenspoek et al. [3] studied the dynamic behavior of active joints for various
electrostatic actuator designs. Hirai et al. [4–6] presented the deflection
characteristics of electrostatic actuators with modified electrode and
cantilever shapes. Wang [7] applied a feedback control for suppressing the
vibration of actuator beams in an electrostatic actuator. Shi et al. [8]
combined an exterior boundary element method for electrostatics and a finite
element method for elasticity to evaluate the coupling effect between the
electrostatic force and the elastic deformation. Gretillat et al. [9] employed three-dimensional
finite element programs to simulate the dynamics of a nonlinear actuator, considering
the effect of squeeze-film damping. Hung and Senturia [10] proposed leveraged
bending and strain-stiffening methods to enlarge the limit of travel distance
before pull-in of electrostatic actuators. This work will analyze the nonlinear
pull-in behaviors of different types 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 differential quadrature method in solving this
problem will be evaluated through a range of case studies. The dynamic
equations of the cantilever microactuator are derived using the differential
quadrature method. The equations describing the residual vibrations of the
microelectrostatic actuators are derived in this paper. The differential
quadrature method is used to produce the electrostatic field equations in
matrix form.


## 2. The Differential Quadrature
Method

This paper employs the differential
quadrature method, with its easy-to-use and meshless technique, to
analyze the nonlinear deflection behaviors of
different types of microactuator with different residual stresses. There are a
number of solution techniques for complicated beam problems, such as the Rayleigh-Ritz
method, the analytical method, the Galerkin method, the finite element method,
and the boundary element method. The differential
quadrature method has been extensively used to solve a variety of
problems in different fields of science and engineering with no need of energy
formulation. The differential quadrature method
has been shown to be a powerful contender in 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 points are
selected at a small distance from each other. The δ technique can be applied to the double
boundary conditions of plate and beam problems. The accuracy of the solution
depends on a sufficiently small δ.
The boundary points are chosen at a small distance δ.
The δ technique can be applied to the double boundary conditions of plate
and beam  problems. The  use of δ at the boundary makes the matrix ill
conditioned [11]. Wang and Bert [12] considered the boundary conditions in finding
the differential quadrature weighting coefficients. Malik and Bert [13] solved
the problem of the free vibration of the plates and showed that the boundary conditions
can be built into the differential quadrature
weighting coefficients. In their
formulation, the multiple boundary conditions are directly applied to the differential quadrature weighting coefficients, and thus it is not necessary to select a nearby
point. In other words, the accuracy of the calculated results will be
independent of the value of the δ-interval. The
differential quadrature weighting
coefficients can be obtained by multiplying the inverse matrix [13]. Sherbourne and Pandey [14] solved buckling problems
using the differential quadrature method. From
the foregoing discussion, over the last two decades, the differential quadrature method has been applied
extensively as an effective means of solving a range of problems in various
fields of science and engineering. Quan and Chang [15, 16] derived the
weighting coefficients in a more explicit way. Feng and Bert [17] analyzed the flexural vibration analysis of a
geometrically nonlinear beam using the quadrature method. Chen and Zhong
[18] presented the study on the nonlinear computations of the differential quadrature method and differential cubature method. Tomasiello [19] applied
the differential quadrature method to
initial-boundary-value problems. Wang et al. [20] presented the free vibration
analysis of circular annular plates with nonuniform thickness by the differential quadrature method. Wang and Gu [21]
presented the static analysis of frame structures by the differential
quadrature element method. Liew et al. [22, 23] presented the differential quadrature method for Mindlin plates on
Winkler foundations and thick symmetric cross-ply laminates with first-order
shear flexibility. Du et al. [24]
presented the application of a generalized differential
quadrature method to structural
problems. Mirfakhraei and Redekop [25] solved the buckling of circular
cylindrical shells using the differential quadrature
method. Moradi and Taheri [26] presented the delamination buckling
analysis of general laminated composite beams using the
differential quadrature method. De Rosa and Franciosi [27] introduced
the exact and approximate dynamic analysis of circular arches using the differential quadrature method. Sun and Zhu [28]
used the upwind local differential quadrature method
for solving incompressible viscous flow. Gu and Wang [29]  presented the free
vibration analysis of circular plates with stepped thickness over a concentric
region by the differential quadrature method. Du et al. [30] presented the generalized differential quadrature method for buckling analysis.  Han and Liew [31] analyzed axisymmetric free vibration of thick
annular plates. Tanaka and Chen [32] applied a dual reciprocity boundaryelement method to transient elastodynamic
problems using the differential quadrature method.
Chen et al. [33] solved the high-accuracy plane stress and plate elements by
the quadrature element method. The essence of the differential
quadrature method is that the derivative of a function at a sample point
can be approximated as a weighted linear summation
of the functional values at all of the sampling points in the domain. Using
this approximation, the differential equation is then reduced to a set of
algebraic equations. The effects of position-dependent electrostatic force and
axial residual stress have all been considered in the proposed models. While the efficiency and accuracy of the Rayleigh-Ritz
method depend on
the number and accuracy of the selected comparison functions; the differential
quadrature method does not have this difficulty of selecting the appropriate comparison
functions. The differential quadrature method approximates the mth order partial
derivative of f(z,t) with respect to z.
For a function f(z,t),
the differential quadrature approximation for the mth order derivative at the ith sampling point is given by (1)∂m∂zm{f(z1,t)f(z2,t)⋮f(zN,t)}≅[Dij(m)]{f(z1,t)f(z2,t)⋮f(zN,t)}fori,j=1,2,…N,  in
which f(zi,t) is the functional value at the sample point zi,
and Dij(m) are the differential quadrature weighting
coefficients of the mth order differentiation attached to these
functional values. Quan and Chang [15, 16] introduced a Lagrangian
interpolation polynomial to overcome the numerical ill conditioning in
determining the differential quadrature weighting coefficients Dij(m),
that is, (2)f(z,t)=∑i=1NM(z)(z−zi)M1(zi)f(zi,t),  where (3)M(z)=∏j=1N(z−zj),M1(zi)=∏j=1,j≠iN(zi−zj)fori=1,2,…,N. Equation (2) is substituted into (1). The differential quadrature
weighting coefficients are then given as (4)Dij(1)=M1(zi)(zi−zj)M1(zj)fori,j=1,2,…,N,i≠j,Dii(1)=−∑j=1,j≠iNDij(1)fori=1,2,…,N.  Once the sampling points, such as zi for i=1,2,…,N,
are selected, the coefficients of the differential quadrature weighting matrix
can be obtained from (4). Higher order derivatives of the differential
quadrature weighting coefficients can also be directly calculated by matrix
multiplication [34], which can be expressed as (5)Dij(2)=∑k=1NDik(1)Dkj(1)fori,j=1,2,…,N,Dij(3)=∑k=1NDik(1)Dkj(2)fori,j=1,2,…,N,Dij(4)=∑k=1NDik(1)Dkj(3)fori,j=1,2,…,N,⋮Dij(m)=∑k=1NDik(1)Dkj(m−1)fori,j=1,2,…,N.  There are many computational methods
available for dynamic analysis. In this paper, the residual vibrations of the
microelectrostatic actuators are investigated using the differential quadrature method. The most convenient approach to solving
a beam structure problem is to uniformly space out the sample points. The
selection of sample points is important for the accuracy of the differential quadrature method solution, but inaccurate
results have been obtained when using this uniform distribution. A nonuniform
sample point distribution, such as the Chebyshev-Gauss-Lobatto distribution [34],
improves the accuracy of the calculation. The integrity and computational
efficiency of the differential quadrature method
in solving this problem will be demonstrated using a set of case studies.
However, an alternative efficient technique is still sought. In this study, the
unequally spaced sample points of each beam using the Chebyshev-Gauss-Lobatto
distribution are chosen as (6)zi=L2(1−cos(i−1)πN−1)fori=1,2,…,N.  The differential
quadrature method has been shown to be a powerful candidate for solving
initial and boundary value problems, and has thus become an alternative to other
methods. The efficiency and the accuracy of Rayleigh-Ritz method
depend on the
number and accuracy of the selected comparison functions, whereas the differential quadrature method does not have such a
difficulty. Like that of any polynomial approach, the
accuracy of the solution using this method is improved by increasing the number
of sample points. The differential quadrature method uses high-order element
level, where the finite element method
approximates a function using low-order polynomials.


## 3. Dynamic Behavior of Microactuators

A
shaped microbeam with a curved electrode is shown in Figure1. The figure
depicts the geometry of a tapered electrostatic microactuator. t0 specifies the thickness at the root 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 a driving voltage is applied
between the fixed-fixed microbeam and the electrode, a position-dependent
electrostatic pressure is distributed to deform the microbeam toward the curved
electrode. The gap between the shaped microbeam and the curved electrode
determines the distribution of the electrostatic 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 improve the electrostatic force
distribution and the deformed shape of the actuator. The kinetic energy of the
microactuator is (7)T=12∫0LρA(∂u∂t)2dz+12∫0LρA(∂v∂t)2dz+12∫0LGJz(∂ϕ∂t)2dz,    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 the twist angle in the direction of the
z-axis, A is the area of the cross-section of the microbeam, Jz is the polar moment in the direction of the z-axis, and ρ is the density of the material of the
actuator.Figure 1
Schematic view of an electrostatic fixed-fixed actuator.While
the external voltagee is applied between the deformable beam and the
fixed electrode, a position-dependent electrostatic pressure is created to pull
the deformable beam toward the ground electrode. This electrostatic pressure is
approximately proportional to the inverse of the square of the gap between
them. When the voltage reaches the critical voltage, the fixed-fixed beam will
be pulled toward the electrode suddenly. The electric fringing effects are
ignored in the following analyses. The strain energy of the microactuator can be approximated
as (8)U=12∫0LE(Iyy(∂2u∂z2)2+2Ixy(∂u∂z)(∂v∂z)+Ixx(∂2v∂z2)2)dz+12∫0LGJz(∂ϕ∂z)2dz−12∫0LP(∂u∂z)2dz−12∫0LP(∂v∂z)2dz,   where E is Young’s modulus of the actuator, G is the shear modulus, and Ixx, Iyy,
and Ixy are the moments of area. The load P is the residual axial loading acting on the
fixed end of the actuator. The value of P is σb0t0. σ is the residual stress, and b0 is the beam width. Because
of the coupling between the mechanical and electrostatic effects, the behavior
of the electrostatic actuator appears more complicated than elastic behavior. The external damping
presents a viscous resistance to transverse displacement of the actuator, and the
internal
damping
provides a viscous resistance to straining of the microactuator material. The
damping forces cu(∂u/∂t), cv(∂v/∂t), and cϕ(∂ϕ/∂t) are assumed for resistance to the transverse
velocity of the actuator. The damping forces cui(∂2/∂z2)(EI(∂3u/∂t∂z2)), cvi(∂2/∂z2)(EI(∂3v/∂t∂z2)), and cϕi(∂/∂z)(GJz(∂2ϕ/∂t∂z)) are assumed for the resistance to the strain
velocity of the microactuator. Considering the electrostatic force and the
internal and external damping effects in the actuator, the virtual work δW done by the bent actuator is (9)δW=−∫0Lcu∂u∂tδudz−∫0Lcui∂2∂z2(EI∂3u∂t∂z2)δudz−∫0Lcv∂v∂tδvdz−∫0Lcvi∂2∂z2(EI∂3v∂t∂z2)δvdz−∫0Lcϕ∂ϕ∂tδϕdz+∫0Lcϕi∂∂z(GJz∂2ϕ∂t∂z)δϕdz+∫0Lε0b0e22(d+S−αsin(πz/L)t0/2−v)2δvdz−∫0Lq(z)δvdz,    where e is the applied voltage, ε0 is the dielectric constant of air, such as ε0=8.85×10−12, b0 is the width of the actuator, and d is the initial gap as shown in Figure 1. The
cross-section area of the actuator is A(z)=b0t0(1+αsin(πz/L)), α is the constant. I(z) is the moment of inertia of the
cross-sectional area of the actuator, which is I(z)=I0(1+αsin(πz/L))3 and I0=b0t03/12.
The shape function S(z) describes the shape of the curved electrode,
and it is presented as S(z)=δe+βsin(πz/L). δe is the fixed end gap 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 the actuator deflection and the electrostatic force, this
residual vibration phenomenon has been studied in only a very few papers, as has
the effect of electrode shape on the residual response. Substituting (7), (8), and (9) into Hamilton’s equation: (10)∫t1t2(δT−δU+δW)dt=0,  the dynamic
deflection of a fixed-fixed micro-actuator can be expressed as the following
nonlinear differential equation: (11)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=ε0b0e22(d+S(z)−αsin(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,  where ε0 is the dielectric constant of air. The
corresponding boundary conditions of the clamped-clamped micro-ctuator are (12)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.  Equation (1) is
substituted into (11)-(12) by employing
the differential quadrature method. The equations of motion of the microactuator
can be discretized in matrix form with respect to the sample points as (13)[M]{∂2w∂t2}+[C]{∂w∂t}+[K]{w}={F}.  The displacement
vector at the sample points is (14){w}={u(z1)u(z2)⋮u(zN)v(z1)v(z2)⋮v(zN)ϕ(z1)ϕ(z2)⋮ϕ(zN)}.The elements in the mass matrix are(15)Mii=0fori=1,2,Mii=ρAfori=3,4,…,N−2,Mii=0fori=N−1,N,Mij=0fori≠j,i=1,2,…,N,j=1,2,…,N,Mij=0fori=1,2,…,N,j=N+1,N+2,…,2N,Mij=0fori=1,2,…,N,j=2N+1,2N+2,…,3N,Mij=0fori=N+1,N+2,…,2N,j=1,2,…,N,Mii=0fori=N+1,N+2,Mii=ρAfori=N+3,N+4,…,2N−2,Mii=0fori=2N−1,2N,Mij=0fori≠j,i=N+1,N+2,…,2N,j=N+1,N+2,…,2N,Mij=0fori=N+1,N+2,…,2N,j=2N+1,2N+2,…,3N,Mij=0fori=2N+1,2N+2,…,3N,j=1,2,…,N,Mij=0fori=2N+1,2N+2,…,3N,j=N+1,N+2,…,2N,Mii=0fori=2N+1,Mii=ρJzfori=2N+2,2N+3,…,3N−1,Mii=0fori=3N,Mij=0fori≠j,i=2N+1,2N+2,…,3N,j=2N+1,2N+2,…,3N. The
elements in the damping matrix are (16)Cij=0fori=1,2,j=1,2,…,3N,Cii=cu+cui∂2∂z2(EIyy)Dii(2)+2cui∂∂z(EIyy)Dii(3)+cuiEIyyDii(4)fori=3,4,…,N−2,Cij=cui∂2∂z2(EIyy)Dij(2)+2cui∂∂z(EIyy)Dij(3)+cuiEIyyDij(4)fori≠j,i=3,4,…,N−2,j=1,2,…,N,Cij=0fori=3,4,…,N−3,N−2,j=N+1,N+2,…,3N,Cij=0fori=N−1,N,j=1,2,…,3N,Cij=0fori=N+1,N+2,…,2N,j=1,2,…,N,Cij=0fori=N+1,N+2,j=N+1,N+2,…,2N,Cii=cv+cvi∂2∂z2(EIxx)Dii(2)+2cvi∂∂z(EIxx)Dii(3)+cviEIxxDii(4)fori=N+3,N+4,…,2N−2,Cij=cvi∂2∂z2(EIxx)Dij(2)+2cvi∂∂z(EIxx)Dij(3)+cviEIxxDij(4)fori≠j,i=N+3,N+4,…,2N−2,j=N+1,N+2,…,2N,Cij=0fori=2N−1,2N,j=N+1,N+2,…,2N,Cij=0fori=N+1,N+2,…,2N,j=2N+1,2N+2,…,3N,Cij=0fori=2N+1,j=1,2,…,3N,Cij=0fori=2N+2,2N+3,…,3N−1,j=1,2,…,2N,Cii=cϕ−cϕi∂∂z(GJz)Dii(1)−cϕi(GJz)Dii(2)fori=2N+2,2N+3,…,3N−1,Cij=−cϕi∂∂z(GJz)Dij(1)−cϕi(GJz)Dij(2)fori≠j,i=2N+2,2N+3,…,3N−1,j=2N+1,2N+2,…,3N,Cij=0fori=3N,j=1,2,…,3N. The elements in the
stiffness matrices are(17)K11=1,K1j=0forj=2,3,…,3N,K2j=D1j(1)Lforj=1,2,…,N,K2j=0forj=N+1,N+2,…,3N,Kij=E∂2Iyy∂z2|z=ziDij(2)+2E∂Iyy∂z|z=ziDij(3)+EIyyDij(4)+PDij(2)fori=3,4,…,N−2,j=1,2,…,N,Kij=E∂2Ixy∂z2|z=ziDi,j−N(2)+2E∂Ixy∂z|z=ziDi,j−N(3)+EIxyDi,j−N(4)fori=3,4,…,N−2,j=N+1,N+2,…,2N,KN−1,j=DN,j−N(1)forj=1,2,…,N,KNj=0forj=1,2,…,N−1,KNN=1,KNj=0forj=N+1,N+2,…,3N,Kij=0fori=3,4,…,N−2,j=2N+1,2N+2,…,3N,Kij=0fori=N+1,j=1,2,…,N,KN+1,N+1=1,Kij=0fori=N+1,i=N+2,N+3,…,3N,Kij=0fori=N+2,j=1,2,…,N,Kij=D1,j−N(1)fori=N+2,j=N+1,N+2,…,2N,Kij=0fori=N+2,i=2N+1,2N+2,…,3N,Kij=E∂2Ixy∂z2|z=ziDi−N,j(2)+2E∂Ixy∂z|z=ziDi−N,j(3)+EIyyDii(4)fori=N+3,N+4,…,2N−2,j=1,2,…,N,Kij=E∂2Ixx∂z2|z=ziDi−N,j−N(2)+2E∂Ixx∂z|z=ziDi−N,j−N(3)+IxxDi−N,j−N(4)fori=N+3,N+4,…,2N−2,i=N+1,N+2,…,2N,Kij=0fori=2N−1,j=1,2,…,N,Kij=D1,j−N(1)fori=2N−1,j=N+1,N+2,…,2N,Kij=0fori=2N−1,j=2N+1,2N+2,…,3N,Kij=0fori=2N+1,j=1,2,…,2N,Kij=1fori=2N+1,j=2N+1,Kij=0fori=2N+1,j=2N+2,…,3N,Kij=0fori=2N+2,2N+3,…,3N,j=1,2,…,2N,Kij=−G∂Jz∂z|z=ziDi−2N,j−2N(1)−GJzDi−2N,j−2N(2)fori=2N+2,2N+3,…,3N−1,j=2N+1,2N+2,…,3N,Kij=0fori=3N,j=1,2,…,3N−1,Kij=1fori=3N,j=3N,Fi=0fori=1,2,…,N+2,Fi=ε0be22(d+S(z)−αsin(πz/L)t0/2−v(z))2−q(z)fori=N+3,N+4,…,2N−2,Fi=0fori=2N−1,2N,…,3N. The dynamic
responses of  the microactuator  are solved  using the Wilson-θ integration method in this paper. The Wilson-θ integration method is an effective implicit time
integration procedure for dynamic problems. It is a step-by-step integration
method that assumes that the acceleration terms vary linearly between consecutive
sampling instants. An electrostatic force pulls the cantilever actuator toward the
curved electrode. The electrostatic force is generated by the difference
between voltage applied to the curved electrode and that applied to the
actuator. This electrostatic pressure is approximately proportional to the
inverse of the square of the gap between them. When the voltage exceeds the
critical voltage, the fixed-fixed beam is suddenly pulled into the electrode.


## 4. Numerical Results and
Discussion

The
microactuator is fabricated from polysilicon material. The geometric parameters and the material
of the microactuator areE=150GPa, δmax=30μm, α=0, β=0, cui=0, cvi=0, cϕi=0, cu=0, cv=0, cϕ=0, b0=5μm, t0=2μm, d=2μm,
and L=500μm.
Figure 2 shows the  deflections of the microbeam
with different applied  voltages. The results  indicate that the results
calculated from the proposed  differential quadrature  method agree very well
with the results found using the finite element method.
Figure 3 shows the frequencies of an electrostatic fixed-fixed actuator for
various lengths of the beam. Again, the results found using the
differential quadrature method are similar to the results found using the
finite element method.
Figure 4 plots the deflections near the middle of an electrostatic
fixed-fixed actuator for various residual stresses. The value of applied
voltageis 620V.
The nonlinear dynamic equation formed by the differential quadrature method is
solved by the Wilson-θ integration method, with θ=1.4 and Δt=0.003 millisecond. A number of papers state that Wilson-θ integration method is unconditionally stable
with a factor of θ≥1.37 [35, 36]. The calculated results show that
higher residual stresses produce smaller deflections near the middle of an electrostatic
fixed-fixed actuator.
Figure 5 shows the stresses near the middle of an electrostatic
fixed-fixed actuator for various residual stresses. Numerical results in this example
show that the residual stresses can significantly affect the
dynamic behavior of the actuator system, showing
that higher residual stresses produce larger stresses near the middle of an electrostatic
fixed-fixed actuator.
Figure 6 shows the stress near the root of an electrostatic
fixed-fixed actuator for various residual stresses. Results indicate that
residual stress is a very sensitive parameter for the residual vibration of the
microactuator. Numerical
results in this example show that the driving
voltage can affect the electromechanical behavior of the actuator system significantly. Calculated results also display that the
higher residual stresses introduce the larger stresses near the root of an electrostatic
fixed-fixed actuator. Residual axial loading should be considered in the
design. Numerical results indicate that the differential quadrature method is a
feasible and efficient method to analyze the nonlinear pull-in behavior of a
fixed-fixed type of electrostatic microbeam.Figure 2
Deflections of an electrostatic fixed-fixed actuator for various applied
voltages.Figure 3
Frequencies of an electrostatic fixed-fixed actuator
for various beam lengths.Figure 4
Deflections near the middle of an electrostatic
fixed-fixed actuator for various residual stresses.Figure 5
Stresses near the middle of an electrostatic
fixed-fixed actuator for various residual stresses.Figure 6
Stresses near the root of an electrostatic
fixed-fixed actuator for various residual stresses.


## 5. Conclusions

The
differential quadrature method is highly suited
to designing or analyzing an electrostatic microactuator. The simplicity of
this formulation makes it a strong candidate for modeling applications that are
more complicated. The effects of residual stresses of microactuators on the
nonlinear pull-in phenomena have also been investigated by employing the
proposed differential quadrature method
algorithm.



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*Source: 905628-2009-03-16.xml*