行政院國家科學委員會專題研究計畫 期中進度報告
微靜電型制動器靜動態響應特性之研究(1/2)
計畫類別： 個別型計畫
計畫編號： NSC93-2212-E-110-020-
執行期間： 93 年 08 月 01 日至 94 年 07 月 31 日
執行單位： 國立中山大學機械與機電工程學系(所)
計畫主持人： 光灼華
計畫參與人員： 陳寶全、張致瑋、張光耀
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報告附件： 出席國際會議研究心得報告及發表論文
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中 華 民 國 94 年 5 月 27 日
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微靜電型致動器靜動態響應特性之研究 (1/2)
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中 華 民 國 九十四 年 五 月
行政院國家科學委員會專題研究計畫 期中報告
微靜電型致動器靜動態響應特性之研究 (1/2)
（期中報告）
計畫編號：NSC 93-2212-E-110-020
執行期限：93 年 8 月 1 日至 94 年 7 月 31 日
主持人：光灼華 中山大學機械與機電工程學系
計畫參與研究生：陳寶全、張致瑋、張光耀 中山大學機械與機電工程學系
Abstract: A two years project is excuted to investigate the
effects of electrode shape and static electricity field on the static and dynamic responses in different electrostatic micro-actuators. The nonlinear electrostatic pull-in behavior for shaped actuators in microelectromechanical systems (MEMS) is simulated in the first phase. The differential quadrature method (DQM) was employed for solving the nonlinear interaction between the curved electrostatic field force and the deflection of shaped cantilever actuators. Different boundary conditions for electrostatically actuated structures, e.g. cantilever beam, fixed-fixed beam, as well as the fringing effect of electrical fields were included in this proposed model. Both the small deflection and large deflection assumptions are implemented in this work to explore the possible effect on accuracy. The numerical results indicate that the DQM can handle this nonlinear actuator problem accurately, efficiently and systematically. The variations of pull-in phenomena at the tip of micro-actuators with different electrode shapes are studied numerically and experimentally.
Keywords: Actuator; curved electrode; electrostatic; large
deflection; MEMS; differential quadrature method
1 INTRODUCTION
Today because of the merits of electrostatic actuators, i.e., good scaling property, lower power driving, large deflection ability, relative ease of fabrication, etc., the electrostatic-actuator applications in microelectromechanical systems (MEMS) have increased tremendously. This is one of the most important devices in these systems.
The fabrication procedure for an electrostatic deflectable metal-coated oxide cantilever beam with a planar electrode is introduced as a light modulator array for micromechanical switches [1]. Two models, i.e. the one-dimensional lumped parallel-plate spring model and the 3D simulation combining with finite-element model were proposed by Osterberg et. al. in 1994 [2] to calculate the pull-in behavior in cantilever structures. The curved electrodes were proposed by Legtenberg et. al. [3] in 1997 for realizing the characteristics of large-displacement actuators. A three-dimensional coupled electromechanical model was proposed to handle the interaction between the actuator deflection and the electrostatic force. The Rayleigh-Ritz method was used to solve the 1D small deflection energy model. The simulated results were compared with the experimental data in this paper. Different designs for the shaped beam and the curved electrode were proposed by Hirai et al. [4-5]. The nonlinear large bending deflection effect was emphasized in the formulation. Furthermore, active joints with a rigid counter-electrode were presented in [6]. Electrostatic forces and their effects on different capacitive mechanical actuators [7] were also investigated.
To extend the travel distance before electrostatic actuator pull-in, the leveraged bending and strain-stiffening methods [8] were proposed. The electrostatic pull-in results from different
actuator structures, e.g. cantilever beams, fixed-fixed beams, and clamped diaphragms, were adopted to extract the Young’s modulus and residual stress in the micro-actuators [9]. Pull-in voltage and capacitance-voltage measurements together with 2-D simulations for electromechanical systems were also performed to extract the material properties of electromechanical structures [10].
Because electrostatic actuated microstructures can undergo large deformation for certain applied voltages, a mixed-regime approach to combine linear and nonlinear theories is proposed [11]. In reference [11],the “finite cloud method” and “a point collocation technique” were employed to formulate and discretize the governing differential equations. In this paper, the differential quadrature method (DQM) is used to formulate the nonlinear deflection equations of different electrostatic actuators into its corresponding discrete forms. The numerical results for different actuator types are simulated and compared with experimental data.
2 ACTUATOR DEVICES
The design of a cantilever shape actuator beam with a curved electrode is shown in Fig. 1. In Fig.2, another type of actuator for fixed-fixed beam suspended above a ground plane is shown. During voltage application to the deformable beam and fixed electrode, a position dependent electrostatic force distribution is created to pull the deformable beam toward the curved electrode. To prevent a short circuit after pull-in contact, an isolation layer or other structure is required.
The shape of the curved electrode s x can be presented as ( ) polynomials and expressed using the following equation: ( )s x d ( )x n
L
δ
= + (1)
where d is the initial gap and δ is the maximum height at 0
x= , L is the beam length and is the polynomial order of the
curved electrode. For different values of , the electrode shape is different, e.g., n
n n 0
= for a rectangular electrode and n 1= for a triangular electrode. Similarly, the shape of the cantilever beam can also be expressed in a polynomial form [4]..
t x( )=t 1o
[
−(x L)]
m (2) where is the thickness at the fixed end and m is thepolynomial order of the cantilever beam. Similarly, the beam shape is rectangular if the polynomial order m is equal to 0.0,
and a triangular beam shape is expected if is equal to 1.0.
o
t
3. DEFLECTION ANALYSIS
When the driving voltage is applied between electrodes, the cantilever beam is deflected by the electrostatic force. Generally, the electrostatic force is approximately proportional to the inverse of the square of the gap between electrodes. For simplicity, a parallel-plate approximation [2] is assumed in this work. After the voltage exceeds the critical voltage, the cantilever beam will be suddenly pulled into the electrode. For evaluating the nonlinear large deflection term and boundary conditions effects in the deflection equation, there are two models for cantilever actuator beam and one model for fixed-fixed actuator beam are considered.
A. Small Deflection Model of a Cantilever Beam Actuator
Based on the small deflection theory of a cantilever beam [2,3], the static deflection of cantilever shaped beam with a distributed transverse load can be expressed as the following nonlinear differential equation.
( ) y x ( , ) p x V
[
]
(
2 2 2 0 2 2 2 ( ) ( ) ( , ) 1 2 ( ) ( ) r bV d d y x EI x p x V F dx dx s x y x ε ⎡ ⎤= = + ⎢ ⎥ − ⎣ ⎦)
(3) (3)where is the Young’s modulus of the beam material, is the moment of inertia of the cross-sectional area,
E I x( )
0 ε is the dielectric constant of air, b is the beam width, ( )s x is the shape
of the curved electrode as function of position x and
[
]
. ( ) ( )
r
F =0 65 s x −d x b is the fringing-field correction [9].
B. Large Deflection Model of a Cantilever Beam Actuator
When a cantilever actuator beam is loaded, its deflection can be described by a relationship between the curvature at any point on the beam and the applied moments at that point [4-5].
1 ( ( ) ) M x EI x ρ = (4) (4)
where ρ is curvature radius of the beam, M x is deflecting ( ) moment acted at point on the beam. The curvature can further be expressed as the following equation with large deflection theory:
(
)
2 2 3 2 2 1 ( ) 1 ( ) d y x dx dy x dx ρ =⎡ ⎤ + ⎣ ⎦ (5) (5)Therefore, the formulation for large bending deflection of shape cantilever beam with curved electrode can be derived as the following nonlinear differential equation.
(
)
[
]
2 2 2 0 3 2 2 2 (1 ) ( ) ( ) 2 ( ) ( ) ( ) 1 ( ) L r x bV F d y x dx L x dx EI x s x y x dy x dx ε + − = − ⎡ + ⎤ ⎣ ⎦∫
i (6)C. Small Deflection Model of a Fixed-Fixed Beam Actuator
The governing differential equations for a fixed-fixed beam type actuator, as shown in Fig. 2, enhanced with first order fringing correction can be presented as [9].
[
]
(
2 2 2 2 0 2 2 2 2 ( ) ( ) ( ) 1 2 ( ) b r bV d d g x d g x EI x T F dx dx dx g x ε ⎡ ⎤− = − + ⎢ ⎥ ⎣ ⎦)
t (7) (7) with Tb=σb ( ) 0 1 σ σ= −υ and Fr=0 65 g x b.[
( )]
where ( )g x is the gap distance, σ and σ0 are the effective
residual stress and the original biaxial residual stress, respectively. υ is Poisson’s ratio. For a beam considered wide,
i.e. , the effective modulus E can be approximated by the plate modulus
b≥5t
/( 2)
E 1−υ , otherwise is equal to Young’s modulus .
E E
4. DIFFERENTIAL QUADRATURE METHOD
In this work, the differential quadrature method (DQM) [12-18] is employed for formulating the nonlinear differential equations mentioned previously into the corresponding discrete forms. The fundamental theory of the DQM is that the different order partial derivatives of a function at a given point can be approximated by a weighted sum of function values at all discrete points in that domain. Considering the m th order derivative of a single function y x( ) at a given point xi can be
expressed by the DQM with N discrete points [17].
( ) 1 ( ) i m N m ij j m j x x y x w y x = = ∂ = ∂
∑
, i=1 2, ,...,N (8) where (yj =y xj), are the corresponding weighting coefficients in DQM. The weighting coefficients of the first order derivatives can be obtained from the following equation. ( )m ij w ( ) N 1 i k ij k i j i j k j k 1 k x x 1 w x x ≠ x x ≠ = − = −∏
− , , ,...,i=1 2 N and j=1 2, ,...,N (9) (1)1
N ii k i i kw
x
x
≠=
−
∑
, i=1 2, ,...,N (10) The weighting coefficients for higher-order derivatives can also be obtained by matrix multiplication [16].( ) ( ) ( ) N 2 1 1 w ij ik kj k 1 w w = =
∑
( ) ( ) ( ) N 3 1 ij ik kj k 1 w w w = =∑
( ) ( ) ( ) N 4 1 3 ij ik kj k 1 w w w = =∑
, 2 , , The Newton-Raphson method is a standard procedure for solving a nonlinear equation set resulting from the DQM for nonlinear differential and integro-differential equations. Hence, it is necessary to introduce the definition for Hadamard product form and SJT product of matrix briefly [16,17]. From these two product methods, the Frechet derivative matrix of Newton-Raphson method can be computed with efficient and explicit procedure.Definition 1. Let matrices A=[aij] and , the
Hadamard product of matrices is defined as . [ ] N M ij B= b ∈C × [ ] N M ij ij A B= a b ∈C × It leads to q [ q] N . ij A = a ∈C ×M
Definition 2. Let matrix [ ] N M and vector
ij
A= a ∈C ×
1
{ } N ij
B= b ∈C × , the SJT product [16,17] of matrix and vector is defined as [ ] N M
ij j
A B◊ = a b ∈C × .
When considering the nonlinear quadratic operatorf f , the ,x ,y
DQM can be expressed with Hadamard product form as
,x ,y ( x ) ( y )
f f = w f w f [17]. Where f is the unknown vector, and
f with the comma x and are the vector of partial derivative of the function along
y
x and direction, respectively. For and
are the weighting coefficient of DQM for the first derivative at
y wx
y
w
x andy directions.
The nonlinear differential equations for small deflection model (3) and large deflection model (6) can be further normalized and
derived using DQM.
A. Small Deflection Model of a Cantilever Beam Actuator:
The equation (3) can be rewritten in dimensionless form as
[
]
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 3m 3 2 3m 3m 4 3 2 2 0 r 2 3 0 d y x d 1 x d y x d 1 x d y x 1 x 2 dx dx dx dx dx 6 V L 1 F Et s x y x ε − − − + + + = − 2 2 ( ) ( ) [ ( )] [ ( )] [ ( )] [ ( )] [ ( )] ... [ ( )] [ ( )] [ ( )] 2 0 r 3 2 3 0 2 3 4 4 5 6 6 V L 1 F 1 2 y x Et s x s x 3 y x 4 y x 5 y x s x s x s x ε + ⎡ = ⎢ + ⎣ ⎤ + + + + ⎥ ⎦ (12) where x=x L, ( )y x =y x L( ) , ( )s x =s x L( ) The corresponding boundary conditions are( ) ( ) 0
y x =y x′ = , at x= 0
and y x′′( )=y′′′( )x = , at x 10 = (13) It should be emphasized that the set of boundary conditions are taken into account in the weighting coefficient matrices. Hence, the weighting coefficient matrices in this case can be shown as follows. ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) , , 1 1 22 2 N 1 2 N 1 ij 1 1 1 N 1 2 N 1 N 1 N 1 N 1 1 N 2 N N 1 N N 0 0 0 0 0 w w w w 0 w w w 0 w w w − − − − − ⎡ ⎢ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , , 1 1 − ⎤ ⎥ 1 , 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ , , 1 1 − (14) ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) ( ) , , , , ( ) ( ) ( ) , , ( ) , 1 1 1 1 11 12 1 N 1 1 N 1 1 1 1 21 22 2 N 1 2 N 2 ij 1 1 1 1 N 1 1 N 1 2 N 1 N 1 N 1 N 1 1 22 2 N 1 2 N 1 N 1 2 N 1 w w w w w w w w w w w w w 0 0 0 0 0 0 0 0 0 w w w 0 w w − − − − − − − − − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⋅ ( ) ( ) , , ( ) ( ) ( ) , , 1 1 N 1 N 1 N 1 1 N 2 N N 1 N N w 0 w w w − − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (15) ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) ( ) , , , , ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) , , 1 1 1 1 11 12 1 N 1 1 N 1 1 1 1 21 22 2 N 1 2 N 3 ij 1 1 1 1 N 1 1 N 1 2 N 1 N 1 N 1 N 1 1 1 1 11 12 1 N 1 1 N 1 1 1 21 22 2 N 1 2 w w w w w w w w w w w w w 0 0 0 0 w w w w w w w w − − − − − − − − − ⎡ ⎢ ⎢ ⎥ ⎢ = ⎢ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⋅ ( ) ( ) ( ) ( ) ( ) , , , , 1 N 1 1 1 1 N 1 1 N 1 2 N 1 N 1 N 1 N w w w w 0 0 0 0 − − − − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢⎣ ( ) ( ) ( ) , , ( ) ( ) ( ) , , ( ) ( ) ( ) , , 1 1 22 2 N 1 2 N 1 1 1 N 1 2 N 1 N 1 N 1 N 1 1 N 2 N N 1 N N 0 0 0 0 0 w w w 0 w w w 0 w w w − − − − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⋅ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (16) ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) ( ) , , , , ( ) ( ) ( ) ( ) , , , , ( ) ( ) ( ) , , 1 1 1 1 11 12 1 N 1 1 N 1 1 1 1 21 22 2 N 1 2 N 4 ij 1 1 1 1 N 1 1 N 1 2 N 1 N 1 N 1 N 1 1 1 1 N 1 N 2 N N 1 N N 1 1 1 11 12 1 N 1 1 w w w w w w w w w w w w w w w w w w w w w − − − − − − − − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ i ( ) ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) , , , , 1 N 1 1 1 1 21 22 2 N 1 2 N 1 1 1 1 N 1 1 N 1 2 N 1 N 1 N 1 N w w w w w w w w 0 0 0 0 − − − − − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) , , , , ( ) ( ) ( ) , , ( ) ( ) , , 1 1 1 1 11 12 1 N 1 1 N 1 1 1 1 21 22 2 N 1 2 N 1 1 1 1 N 1 1 N 1 2 N 1 N 1 N 1 N 1 1 22 2 N 1 2 N 1 1 N 1 2 N 1 N 1 N 1 w w w w w w w w w w w w 0 0 0 0 0 0 0 0 0 w w w 0 w w w − − − − − − − − − − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ i ( ) , ( ) ( ) ( ) , , 1 N 1 1 N 2 N N 1 N N 0 w w − w 1 − , 1 ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (17)
Furthermore, the basic iteration equation can be written using the differential quadrature method.
( ) ( ) ( ) ( ) ( ) { } ( ) 3m 2 3m 3m 4 3 2 ij ij 2 ij d 1 x d 1 x y 1 x w y 2 w y w dx dx ϕ = − + − + − y ( ) [ ] [ ( )] [ ( )] [ ( )] [ ] [ ] [ ( )] [ ( )] 2 2 0 r 3 2 3 4 0 3 4 5 6 6 V L 1 F 1 2 y 3 y Et s x s x s x 4 y 5 y s x s x ε + ⎡ − ⎢ + + ⎣ ⎤ + + + ⎥ ⎦ (18) ( ) ( ) ( ) { } ( ) ( ) ( ) 3m 2 3m 3m 4 3 2 ij ij 2 ij y d 1 x d 1 1 x w 2 w w y dx ϕ ∂ = − + − + ∂ x dx − ( ) [ ] [ ( )] [ ( )] [ ] [ ] + [ ( )] [ ( )] 2 0 r u u 3 3 4 0 2 3 u u 5 6 6 V L 1 F 2I 6 I y Et s x s x 12I y 20I y s x s x ε + ⎡ ◊ − ⎢ + ⎣ ⎤ ◊ + ◊ + ⎥ ⎦ (19)
where
I
u is a unit matrix.From equations (18) and (19), the iteration formulation of Newton-Raphson method for this case is
( ) ( ) ( ) ( ) { } { } k k 1 k k y y y y y ϕ ϕ + = − ∂ ∂
The linear solution of a cantilever beam with a uniform distribution load is chosen to be the initial guess in the iteration procedure.
B. Large Deflection Model of a Cantilever Beam Actuator:
Similarly, the equation (6) can be rewritten as
(
)
( ) ( ) ( ) ( ) 2 2 2 0 r 3 3 3m 2 2 0 6 V L 1 F d y x dx Et 1 x 1 dy x dx ε + = − ⎡ + ⎤ ⎣ ⎦[
]
[
]
( ) ( ) ( ) ( ) ( ) ( ) 1 x 2 2 0 0 1 x 1 x dx dx s x y x s x y x ⎧ − − ⎪ − ⎨ ⎬ − − ⎪ ⎪ ⎩∫
∫
⎭ i ⎫⎪ (21) Equation (21) can further be expanded using the Taylor series as follows. ( ) ( ) ( ) ( ) ... 2 4 6 2 2 d y x 3 dy x 15 dy x 35 dy x 1 dx 2 dx 8 dx 16 dx ⎧ ⎫ ⎪ − ⎡ ⎤ + ⎡ ⎤ − ⎡ ⎤ ⎪ ⎨ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎬ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎪ ⎪ ⎩ ⎭ i +[
]
[
]
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 0 r 3 3m 0 1 x 2 2 0 0 6 V L 1 F Et 1 x 1 x 1 x dx dx s x y x s x y x ε + = − ⎧ − − ⎫ ⎪ − ⎪ ⎨ ⎬ − − ⎪ ⎪ ⎩∫
∫
⎭ i (22)In terms of differential quadrature method, the basic iteration equations for Newton-Raphson method are shown as follows.
( ) ( ) ( ) ( ) ( ) ( ) ( ) { } [ ] [ ] [ ] ... 2 2 1 2 2 1 ij ij ij ij ij 2 1 6 ij ij 3 15 y w y w y w y w y w y 2 8 35 w y w y 16 ϕ = − + 4 − + ( ) ( ) ( ) [ ( )] [ ( )] [ ( )] [ ( )] [ ( )] [ ( )] [ ( )] [ ( )] 2 N 0 r k k k 3 2 k 1 0 k 2 3 4 k k k 4 5 6 k k k 6 V L 1 F 1 2 y x C 1 x Et s x s x 3 y x 4 y x 5 y x s x s x s x ε = ⎡ + − − •⎢ + ⎣ ⎤ + + + 3 k + ⎥ ⎦
∑
( ) ( ) [ ( )] ( ) [ ( )] [ ( )] [ ( )] 2 N 0 r k k 3 2 k 1 0 k 2 k k 3 4 k k 6 V L 1 F 1 x C 1 xx Et s xx 2 y xx 3 y xx s xx s xx ε = ⎡ + + − • ⎢ ⎣ + +∑
[ ( )] [ ( )] ... ... [ ( )] [ ( )] 2 3 k k 3 4 k k 12 y xx 20 y xx s xx s xx ⎤ + + + ⎥ ⎦ (23)where is derived by the Newton-Cotes integration formulas [18]. k C N 1 i k 0 i 1 k i i k x x C dx x x = ≠ − = −
∏
∫
(24) It leads to ( ) ( ) ( ) ( ) ( ) ( ) { } [ ] [ ] 2 1 2 2 1 ij ij ij ij ij ij y 3 w w y w 3 w y w w y 2 ϕ ∂ = − ◊ − ◊ ◊ ∂ 1 y ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ 1 ]4 2 [ 2 ] 1 [ 1 ]3 [ 1 ]6 2 ij ij ij ij ij ij ij 15 15 35 w y w w y w w y w y w 8 2 16 + ◊ + ◊ ◊ − ◊ ( ) ( ) ( ) ( ) [ ] [ ] ( ) [ ( )] [ ( )] [ ( )] ... [ ( )] [ ( )] [ ( )] [ ( )] 2 N 2 1 1 5 0 r ij ij ij 3 k k k 1 0 2 u u k u k u k 3 4 5 6 k k k k 6 V L 1 F 105 w y w w y C 1 x 8 Et 2I 6 I y x 12I y x 20 I y x s x s x s x s x ε = + − ◊ ◊ − − ⎡ ◊ ◊ ◊ 3 •⎢ + + + + ⎣∑
⎤ ⎥ ⎦ ( ) [ ( )] ( ) [ ( )] [ ( )] 2 N 0 r u u k k k 3 3 k 1 0 k 6 V L 1 F 2I 6 I y xx x C 1 xx Et s xx s xx ε = ⎡ + 4 k + − •⎢ + + ⎣∑
◊ [ ( )] [ ( )] ... [ ( )] [ ( )] 2 3 u k u k 5 6 k k 12I y xx 20 I y xx s xx s xx ⎤ ◊ + ◊ + ⎥ ⎦ (25) The deflection of the cantilever beam can be calculated using the above iteration formula for the Newton-Raphson method.C. Small Deflection Model of a Fixed-Fixed Beam Actuator::
Similar to the cantilever actuator procedure, the boundary
conditions of a fixed-fixed beam are ( ) ( )
y x =y x′ = , at x 00 =
and ( )y x =y x′( )= , at x 10 = (26) Hence, the weighting coefficient matrices of DQM in terms of fixed-fixed boundary conditions can be derived as follows.
( ) ( ) , ( ) ( ) ( ) , , 1 1 22 2 N 1 1 ij 1 1 N 1 2 N 1 N 1 0 0 0 0 0 w w 0 w 0 w w 0 0 0 0 0 − − − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) ( ) ( ) , , , , ( ) ( ) ( ) ( ) , , , , 1 1 1 1 11 12 1 N 1 1 N 1 1 1 1 21 22 2 N 1 2 N 2 1 ij ij 1 1 1 1 N 1 1 N 1 2 N 1 N 1 N 1 N 1 1 1 1 N 1 N 2 N N 1 N N w w w w w w w w w w w w w w w w w w − − − − − − − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ = • ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) ( ) ( ) , , , , ( ) ( ) ( ) ( ) , , , , 1 1 1 1 11 12 1 N 1 1 N 1 1 1 1 21 22 2 N 1 2 N 3 2 ij ij 1 1 1 1 N 1 1 N 1 2 N 1 N 1 N 1 N 1 1 1 1 N 1 N 2 N N 1 N N w w w w w w w w w w w w w w w w w w − − − − − − − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ = • ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , and ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) ( ) ( ) , , , , ( ) ( ) ( ) ( ) , , , , 1 1 1 1 11 12 1 N 1 1 N 1 1 1 1 21 22 2 N 1 2 N 4 3 ij ij 1 1 1 1 N 1 1 N 1 2 N 1 N 1 N 1 N 1 1 1 1 N 1 N 2 N N 1 N N w w w w w w w w w w w w w w w w w w − − − − − − − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ = • ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (27)
Following a procedure similar to the cantilever actuator, the basic iteration equations for Newton-Raphson method in terms of differential quadrature method can also be obtained for fixed-fixed beam actuated systems.
5. NUMERICAL RESULTS
To present the feasibility of using the differential quadrature method (DQM) in solving the nonlinear actuator deflection problem, the experimental data and results predicted by employing the energy model (2-D simulation) and MEMCAD (3-D simulation [3,9]) are compared with the results calculated from the proposed algorithm.
A. Cantilever Rectangular Beam Actuators with Different
Shape Electrodes
The parameters of different combinations of cantilever beam and curved electrode are listed in Table I. The Young’s modulus of the beam material, i.e. polysilicon, is E =150 GPa. The maximum height is δ= 30 µm, the width of beam is b = 4.6
m
µ and the length of beam is L = 500 mµ [3]. The calculated and the measured pull-in voltages for these beam actuators are listed in Table II. The variations in tip deflections for different beam and electrode combinations are presented in Figures 3 to 5. The tip deflections of a rectangular beam (m = 0) with a rectangular electrode (n=0) under different driving voltages are shown in Fig. 3. The tendency of tip deflection variation between the measured data and the results calculated from the proposed DQM algorithm is similar. However, as the calculated results provided in [3] show, a significant difference between the predicted and the measured pull-in voltages are found in this analysis. A similar result, as shown in Fig. 4, is observed for a rectangular beam (m=0) with a triangular electrode (n=1). In Fig.
5, the fringing field effect was considered. As mentioned in [3], the ground plane will significantly reduce the fringing field contribution. Therefore, only half of the fringing field was considered in this analysis. The results in Table II indicate that the results calculated using the proposed DQM algorithm agree well with the energy model results [3]. In the proposed DQM algorithm, only twelve sampling points on the beam are used, and the convergence with a tolerance of 10-8 can be achieved within four iterations. The convergent rate of tip deflection calculated using DQM with a different number of selected sampling points is compared in Table III. The locations of sampling points on the beam in this study are selected by employing the so-called Chebyshev-Gauss-Lobatto distribution, which is defined as [15]. cos ( ) ( ) i 1 i 1 x 1 2 N 1 π ⎡ − ⎤ = ⎢ − ⎥ − ⎣ ⎦, for i=1 2, ,...,N (28) The results in Figures 3 to 4 indicate that the difference
between the calculated and the measured pull-in voltages is significant for the case with a uniform rectangular electrode. This might be explained by the 3-D electrostatic field effects and the ground plane presence [3].
B. Difference Introduced from Using the Large and Small
Deflection Models
To improve the significant difference between the measured and the calculated results in uniform rectangular beam actuator ( ) systems, the complicated large deflection beam theory was applied instead of the small beam deflection theory. However, the numerical results, as shown in Fig. 6, show that using the large beam deflection theory cannot improve the difference in the predicted pull-in voltages in these analyzed cases. The results in Fig. 6 show that a significant difference between the pull-in voltages calculated from the large and the small deflection theories can only be found for the cases with
m= =n 0
L=100 mµ , .
t=1 6 mµ and the gap d> 15 mµ . For the actuators with a beam length larger than 300 mµ , the pull-in voltages calculated from the small deflection and the large deflection theories are almost identical. Therefore, it can be concluded that it is necessary to adopt the large deflection theory only when the beam length is smaller than 100 mµ and the gap is larger than 15 mµ . A similar conclusion was found in [11]. However, several hundred sampling points on the beam are necessary for the finite cloud method presented in [11]. For DQM, only twelve sampling points and four iterations were necessary to reach convergence even when applying the complicated large deflection model.
C. Fixed-Fixed Rectangular Beam Actuators with Different
Residual Stresses
To present the DQM performance for fixed-fixed beam actuators, results calculated using the proposed DQM, the 2-D distributed model and the full 3-D MEMCAD self-consistent electromechanical simulation proposed in reference [9] were compared. Six fixed-fixed beam actuator systems with different residual stress distributions were simulated and compared in Table IV. The parameters of the actuator systems [9] are: E=169 GPa,
υ=0.06,
b
=50µm,t
=3µmand the initial gap g =1µm. Results in Table IV indicate that pull-in voltages calculated from DQM agree well with the results calculated from the 2-D distributed model and the 3-D MEMCAD model [9]. In the 2-D distributed model [9], 100 discrete points on the beam are required to converge the pull-in voltage using the finite difference iterative relaxation technique. However, for DQM only 19 sampling points were required to converge the calculated pull-in voltages within a tolerance of 10-8 in four iterations.D. Different Combinations of Shaped Beam and Curved Electrode
As noted in references [4] and [5], different cantilever beam shapes with different curved electrodes may require different driving voltages to achieve a specific beam tip deflection. In this study, the pull-in voltages of the shaped cantilever beam and curved electrode was also calculated. The cantilever shape beam was assumed to be trapezoid and changed gradually from a rectangle into a triangle. The electrode shape is considered to be triangular and parabolic (i.e. n=1&2). The length of the beam is
500 mµ , thickness at fixed end is t0=1 7 m. µ , is the thickness of cantilever at free end and the maximum gap is
e
t
.
33 2 m
δ = µ . The variation in pull-in voltage for the combinations of different shaped beams and curved electrodes are shown in Fig. 7. The results show that the pull-in voltage decreases gradually as the cantilever beam shape alternates from rectangular to triangular (i.e. t te o= to 1 t te o= ). When the electrode is modified into a 0
parabolic shape, the required driving voltage for a specified beam tip deflection can be further decreased. Hence, it is evident that the triangular shape cantilever with 2nd order polynomial curved electrode presents an excellent structure to achieve large deflection with lower applied voltage. Similar results were also reported in [4] and [5].
The results in this work indicate that the proposed DQM approach is not only an alternative but also an efficient numerical method for solving the complicated nonlinear electrostatic behavior for shaped electrode actuators
6. CONCLUSIONS
The primary contributions of this project are as follows. 1. A numerical algorithm using the differential quadrature
method was proposed for electrostatic actuators with shaped cantilever and curved electrode. Fewer sampling points and Newton-Raphson iterations are required to achieve convergence for different types beam actuators.
2. The proposed DQM algorithm can be applied to different shaped beam actuators with good accuracy. Results in the cantilever and fixed-fixed beam actuated structures indicate that the proposed DQM algorithm is more efficient than other methods for analyzing the complicated nonlinear electrostatic behavior for shaped electrode actuators.
3. The difference between the results solved using the small and the large deflection theories shows that the complicated large deflection theory model in the cantilever beam type actuator analysis is not necessary. In general, the pull-in voltages calculated using these two models are nearly identical when the length of the cantilever is larger than 300 mµ and the gap is smaller than 15 mµ .
4. The results show that the triangular cantilever actuator with a 2nd order polynomial curved electrode presents an excellent structure for achieving large deflection with the lower applied voltage.
ACKNOWLEDGEMENTS
The authors would like to acknowledge the support of the National Science Council, Taiwan, R. O. C., through grant no. NSC 93-2212-E-110-020.
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Fig.1. Schematic view of a shaped cantilever and curved electrode micro-actuator.
Fig. 2. Schematic cross section of a fixed-fixed pull-in electrostatic micro-actuator.
Fig.3. Tip deflections for the rectangular beam with rectangular electrode (m = 0, n = 0).
Fig.4. Tip deflections for a rectangular beam with a triangular electrode (m = 0, n = 1).
Fig.5. Tip deflections for a rectangular beam with a parabolic electrode (m=0,n=2).
Fig.6. The pull-in voltages calculated using the small and large deflection theories (t=1 6 m. µ ,m= =n 0).
Fig.7. Pull-in voltages for trapezoid cantilever beam actuators ( m=1 ,n=1&2).
TABLE I The parameters of polysilicon cantilever beams and curved electrodes
Parameters of
Cantilever Beam and Curved Electrode Shape Order (m, n) Thickness of Beam [ o t m µ ] Initial Gap d [ mµ ] (0,0) 1.6 2.2 (0,1) 1.7 3.2 (0,2) 1.8 3.3
TABLE II The pull-in voltages for different polysilicon cantilever beam actuators
Pull-in Voltage [V] Cantilever Beam Order
(m, n)
Proposed Measured Data [3] Energy Model [3]
(0,0) 132 223 112 (0,1) 75 98 74 (0,2) 43.4 40 37
TABLE III Tip deflections of cantilever beam actuator solved with different numbers of sampling points in the DQM algorithm
Tip Deflection (dimensionless by beam length) Solved by DQM
Number of Selected Sampling Points* Order (m, n) 6 8 10 12 (0,0) 0.0228951 0.0228822 0.0228821 0.0228821 (0,1) 0.0115111 0.0112230 0.0111751 0.0111652 (0,2) 0.0127913 0.0127116 0.0127681 0.0127635 (1,0) 0.0072416 0.0067802 0.0067510 0.0067490 (1,1) 0.0221470 0.0204561 0.0203513 0.0203405 (1,2) 0.0106007 0.0105426 0.0105484 0.0105469 Sample points were selected using the
Chebyshev-Gauss-Lobatto distribution and for each case with their own voltage to calculate tip deflection.
TABLE IV Difference in the calculated pull-in voltages for different fixed-fixed beam actuators using DQM, 2-D distributed model and MIT MEMCAD [9] models
Beam Properties Length (µm) Stress (MPa)
VPI (DQM) VPI (2-D) [9] VPI (MEMCAD) [9] 1(%) ∆ ∆2(%) [9] 250 0 39.8 39.5 40.1 0.8% 1.5% 250 100 57.4 56.9 57.6 0.4% 1.2% 250 -25 33.7 33.7 33.6 0.3% 0.3% 350 0 20.5 20.2 20.3 1.0% 0.5% 350 100 36.0 35.4 35.8 0.6% 1.1% 350 -25 13.82 13.8 13.7 0.9% 0.7% 1 ( ) (%) ABS MEMCAD DQM MEMCAD − ∆ = 2 ( 2 ) (%) ABS MEMCAD D MEMCAD − ∆ = , [9]
Convergent Rate of DQM (fixed-fixed beam) NO. of
Sampling Points
13 15 17 19 Center Deflection*
(
×
10
−3) 0.537255 0.537175 0.537213 0.537212 *Center deflection is dimensionless by beam length.第一階段計畫結果自評： 第一階段之計劃，基本上已依原計劃構想執行完畢.。 在本年度計畫中建構了微型靜電制動器之驅動模式， 並經由參數分析，探討不同電極形狀對其非線性 Pull-in 效應之影響，分析結果顯示，適當的設計電極與制動 樑，可有效的提高其 Pull-in 制動電壓與線性範圍，本 計畫提出之分析與設計模式，非僅可改善現有微型靜 電制動器之工作範圍達一倍以上，更提供了一個有效 之設計與分析模式，相關結果均予實驗數據極為吻 合，此點亦顯示所提模式之廣泛適用性與精度。相關 研究結果亦已整理完成，目前已投稿 Int. J. of Mechanical Science 與 J. of Microelectronmechanics 期刊 尋求發表中。
