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Dynamic Analysis of Electrostatic Microactuators Using the Differential Quadrature Method

Dynamic Analysis of Electrostatic Microactuators Using the Differential Quadrature Method Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 576079, 8 pages doi:10.1155/2011/576079 Research Article Dynamic Analysis of Electrostatic Microactuators Using the Differential Quadrature Method Ming-Hung Hsu Department of Electrical Engineering, National Penghu University of Science and Technology, Penghu, Taiwan Correspondence should be addressed to Ming-Hung Hsu, hsu.mh@msa.hinet.net Received 3 January 2011; Accepted 26 April 2011 Academic Editor: Andrew Fleming Copyright © 2011 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. This work studies the dynamic behavior of electrostatic actuators using finite-element package software (FEMLAB) and differential quadrature method. The differential quadrature technique is used to transform partial differential equations into a discrete eigenvalue problem. Numerical results indicate that length, width, and thickness significantly impact the frequencies of the electrostatic actuators. The thickness could not affect markedly the electrostatic actuator capacities. The effects of varying actuator length, width, and thickness on the dynamic behavior and actuator capacities in electrostatic actuator systems are investigated. The differential quadrature method is an efficient differential equation solver. 1. Introduction model, the one-dimensional numerical model, and the finite-element model that incorporates a three-dimensional Plate-type electrostatic actuators are widely applied in simulation, were proposed to calculate the pull-in behaviors microelectromechanical systems. Microelectrostatic actua- of various fixed-fixed DMD structures and pressure sensors tor devices have a high operating frequency, low-power [8]. Gilbert et al. [9] analyzed the three-dimensional coupled consumption and can replace many passive components. electromechanics of MEMS using a CoSolve-EM simulation Mehdaoui et al. [1] presented the vertical cointegration of algorithm. Elwenspoek et al. [10] studied the dynamic AlSi MEMS tunable capacitors and Cu inductors for tunable behavior of active joints for various electrostatic actuator LC blocks. Etxeberria and Gracia [2] presented tunable designs. Shi et al. [11] presented a combination of an exterior MEMS volume capacitors for high-voltage applications. Liu boundary element method for electrostatics combined with et al. [3] presented that actuation by electrostatic repulsion a finite-element method for elasticity to evaluate the effect is produced by nonvolatile charge injection. Gallant and of coupling between the electrostatic force and the elastic Wood [4] investigated how fabrication techniques affect the deformation. Osterberg and Senturia [12] applied the sharp performance of widely tunable micromachined capacitors. instability phenomena of electrostatic pull-in behaviors for Borwick III et al. [5] analyzed a high Q, large tuning cantilever beam and fixed-fixed beam actuators to elicit range MEMS capacitor for RF filter systems. Harsh et al. the material characteristics of MEMS. Gretillat et al. [13] [6] studied the realization and design considerations of a employed three-dimensional MEMCAD and FEM programs flip-chip integrated MEMS tunable capacitor. Petersen [7] to simulate the dynamics of a nonlinear actuator, considering first described the nonlinear pull-in behavior of an electro- the effect of squeeze-film damping. Hung and Senturia [14] static microactuator. Osterberg et al. [8] proposed different developed leveraged bending and strain-stiffening methods numerical models for analyzing electrostatically deformed to increase the limiting travel distance before pull-in of diaphragms. Their results revealed that the electrostatic electrostatic actuators. Chan et al. [15] measured the pull-in deformation calculated using the one-dimensional model voltage and capacitance-voltage and performed 2D simula- is close to that obtained using a three-dimensional model. tions that included the electrical effects of fringing fields and Various models, including the lumped parallel-plate spring finite-beam thickness to determine the material properties of 2 Advances in Acoustics and Vibration 1.4E − 12 1.2E − 12 1E − 12 8E − 13 6E − 13 4E − 13 2E − 13 0E +0 02468 10 12 14 Electrode −5 ×10 176 micrometer 236 micrometer 196 micrometer 256 micrometer Figure 1: Schematic view of electrostatic actuator. 216 micrometer 276 micrometer Figure 3: Capacities of microelectrostatic actuator for various 1.2E − 12 widths. 1E − 12 8E − 13 6E − 13 1E − 12 9E − 13 4E − 13 8E − 13 2E − 13 7E − 13 0E +0 6E − 13 5E − 13 02468 10 12 14 4E − 13 −5 ×10 3E − 13 2E − 13 1E − 13 262 micrometer 322 micrometer 0E +0 282 micrometer 342 micrometer 02468 10 12 14 −5 302 micrometer 362 micrometer × 10 Figure 2: Capacities of microelectrostatic actuator for various 38 micrometer 98 micrometer lengths. 58 micrometer 118 micrometer 78 micrometer 138 micrometer Figure 4: Capacities of microelectrostatic actuator for various gap electrostatic microactuators. Li and Aluru [16] developed a distances. mixed-regime approach for combining linear and nonlinear theories to analyze large MEMS deformations at large applied voltages. Chyuan et al. [17–19] established the validity and partial differential equations and integral equations. The accuracy of the dual boundary element method and applied convergence and accuracy of the finite-element solution it to study the effect of gap size variation for the levitation of depends on the differential equation, integral form, and MEMS comb drive. Lai and Chen [20] studied the influence element used. The energy stored in the microdevice is of the holes in the membrane structures of radiofrequency expressed as follows [22–24]: MEMS switches. Qiao et al. [21] presented the suspension beam called two-beam to achieve parallel-plate actuator with W = ρ V dv,(1) e v extended working range, but without penalties of complex control circuit and large actuation voltage. In this work, where ρ is the volume charge density, v is the volume, and the finite-element method and the differential quadrature V is the applied voltage: method are employed to analyze how actuator length, width, and thickness affect dynamic behavior and capacitances in ρ = ∇· D,(2) v e electrostatic actuator systems. where D is the vector of electric flux density, 2. Capacitance of Electrostatic Micro Actuators 2W C =,(3) Figure 1 depicts the geometry of an electrostatic actuator where h is the thickness of the movable plate [22]. The length where C is the capacitance. The capacitance models are and width of the movable plate are a and b,respectively. extracted using three-dimensional simulations. Figure 2 The microactuator design is based upon the deformation of shows the capacitances of the electrostatic microactuator a movable mechanical structure by electrostatic forces using with various lengths. The width, thickness, and gap distance a fixed electrode. A voltage applied across the gap creates of the electrostatic actuator are 176, 8, and 38 micrometers, the electrostatic force. Electrostatic actuator devices are fabri- respectively. The computational solution is acquired by the cated from polysilicon by surface micromaching techniques. finite-element scheme. The FEMLAB finite-element package In this work, the finite-element technique is applied to find is used to analyze the model. In previous cases, 62463 solid the capacitance of the electrostatic microactuator. The finite- element method is used to find approximate solutions for elements are used to calculate the capacitance of the actuator Farad C (Farad) C (Farad) Advances in Acoustics and Vibration 3 9.08E − 14 where δW is the virtual work. This leads to the following 9.07E − 14 equations for electrostatic actuator motion: 9.06E − 14 9.05E − 14 9.04E − 14 4 4 ∂ w x, y, t ∂ w x, y, t 9.03E − 14 D +2D 9.02E − 14 4 2 2 ∂x ∂x ∂y 9.01E − 14 (7) 0 2 4 6 8 10 12 14 16 18 4 2 ∂ w x, y, t ∂ w x, y, t −5 ×10 V + D + ρh = 0, 4 2 ∂y ∂t 8 micrometer 14 micrometer 16 micrometer 10 micrometer 12 micrometer The boundary conditions of the actuator are as follows: Figure 5: Capacities of microelectrostatic actuator for various thicknesses. w(0, 0, t) = 0, w(a,0, t) = 0, and provide convergent results. The effect of the actuator ( ) w 0, b, t = 0, length is more pronounced at higher capacity. Figure 3 shows the capacitances of the electrostatic microactuator for w(a, b, t) = 0, varying widths. The length, thickness, and gap distance of 2 2 ∂ w x, y, t ∂ w x, y, t the electrostatic actuator are 262, 8, and 38 micrometers, + ν = 0, 2 2 respectively. Numerical results reveal that the capacity of the ∂x ∂y actuator increases as the width of the actuator increases. for x = 0, 0 <y < b, Figure 4 shows the capacitances of electrostatic microac- 3 3 ∂ w x, y, t ∂ w x, y, t tuator for various gap distances. The width, length, and + (2 − ν) = 0, 3 2 ∂x ∂x ∂y thickness of the electrostatic actuator are 176, 262, and for x = 0, 0 <y < b, 8 micrometers, respectively. Numerical results reveal that the capacity of the actuator increases as the gap distance 2 2 ∂ w x, y, t ∂ w x, y, t of the actuator falls. Figure 5 shows the capacitances of + ν = 0, 2 2 the electrostatic microactuator at various thicknesses. The ∂x ∂y width, length, and gap distance of the electrostatic actuator for x = a,0 <y < b, are 176, 262, and 38 micrometers, respectively. Numerical (8) 3 3 results show that the thickness does not significantly affect ∂ w x, y, t ∂ w x, y, t + (2 − ν) = 0, 3 2 the capacitances of the electrostatic microactuator. The ∂x ∂x ∂y evaluation illustrates capacitance problems. for x = a,0 <y < b, 2 2 ∂ w x, y, t ∂ w x, y, t 3. Vibration Analysis of Electrostatic + ν = 0, 2 2 ∂y ∂x Micro Actuators for 0 <x < a, y = 0, The electrostatic actuator has length a, width b, and thickness 3 3 ∂ w x, y, t ∂ w x, y, t h. The strain energy of the actuator is [25] + (2 − ν) = 0, 3 2 ∂y ∂x ∂y for 0 <x < a, y = 0, b a 2 2 1 ∂ w x, y, t ∂ w x, y, t U = D + dx dy,(4) 2 2 2 2 2 ∂x ∂y ∂ w x, y, t ∂ w x, y, t 0 0 + ν = 0, 2 2 ∂y ∂x where w is the deflection of the actuator, D = Eh /(12(1 − for 0 <x < a, y = b, ν )) is the flexural rigidity, E is Young’s modulus, ν is the 3 3 Poisson’s ratio, and h is the actuator thickness. The kinetic ∂ w x, y, t ∂ w x, y, t ( ) + 2 − ν = 0, 3 2 energy of the microactuator is ∂y ∂x ∂y for 0 <x < a, y = b, b a 1 ∂w x, y, t T = ρh dx dy,(5) 2 ∂t 0 0 iωt After substituting w(x, y, t) = W(x, y)e into (7), (7)can be rewritten as follows: where t is the time and ρ is the density of the actuator material. Equations (4)and (5) are substituted into Hamilton 4 4 4 equation as follows: ∂ W x, y ∂ W x, y ∂ W x, y D +2D + D 4 2 2 4 ∂x ∂x ∂y ∂y (9) (δT − δU + δW)dt = 0, (6) = ω ρhW x, y , C (Farad) 4 Advances in Acoustics and Vibration where ω is the natural frequency of the actuator. The stress-strain relationship for linear conditions is as follows: boundary conditions of the electrostatic actuator are as follows: [σ] = D [ε], (12) where [D] is an elastic matrix. The variables are approxi- ( ) W 0, 0 = 0, mated with functions in the chosen finite-element spaces. W(a,0) = 0, The following finite-element solution is assumed [23, 24]: W(0, b) = 0, ∗ ∗ w = N [W ] cos(ωt), (13) W(a, b) = 0, where [N] is the matrix of any suitable assumed shape 2 2 function and [W ] is the displacement matrix. The finite ∂ W x, y ∂ W x, y + ν = 0, element assembles all elements to form a complete structure 2 2 ∂x ∂x to equilibrate a structure with its environment. The equation for x = 0, 0 <y < b, for the finite-element model of the electrostatic microactua- 3 3 ∂ W x, y ∂ W x, y tor is as follows: + (2 − ν) = 0, 3 2 ∂x ∂x ∂y ∗ ∗ 2 ∗ ∗ [K ][W ] = ω [M ][W ], (14) for x = 0, 0 <y < b, ∗ ∗ 2 2 where [M ] is the mass matrix and [K ] is the stiffness ∂ W x, y ∂ W x, y + ν = 0, matrix. The stiffness matrix can be written as follows: 2 2 ∂x ∂y for x = a,0 <y < b, T [K ] = L N D L N dV , (15) (10) 3 3 ∂ W x, y ∂ W x, y + (2 − ν) = 0, 3 2 ∂x ∂x ∂y where [L] is a linear differential operator matrix. The mass for x = a,0 <y < b, matrix can be written as follows: 2 2 T ∂ W x, y ∂ W x, y ∗ [M ] = N ρ N dV. (16) + ν = 0, 2 2 ∂y ∂x for 0 <x < a, y = 0, ∗ ∗ ∗ The matrix order is N × N ,where N is the number 3 3 of nodes for which the solution is unknown. Assembly of ∂ W x, y ∂ W x, y + (2 − ν) = 0, all element stiffness matrices and element mass matrices of 3 2 ∂y ∂x ∂y the electrostatic actuator defines the following eigenvalue for 0 <x < a, y = 0, equation: 2 2 ∂ W x, y ∂ W x, y ∗ 2 ∗ + ν = 0, [K ] − ω [M ] = 0. (17) 2 2 ∂y ∂x for 0 <x < a, y = b, The eigenvalues of the electrostatic actuator can be derived from (17) and are known as eigenfrequency problems. 3 3 ∂ W x, y ∂ W x, y ( ) + 2 − ν = 0, 3 2 ∂y ∂x ∂y 5. Differential Quadrature Formula for 0 <x < a, y = b, The vibration response of the microactuator is numerically 4. Finite-element Method Model modeled using the differential quadrature method in this work. The differential quadrature method is used to convert The commercially available FEMLAB software package is the partial differential equations of the plates into a discrete used to evaluate dynamic problems based on partial differ- eigenvalue problem. The roots of shifted Chebyshev and ential equations. To derive finite formulations, the following Legendre sampling point equation are used to select the sam- virtual work principle must be utilized in the following pling points in these analyses. The integrity and computa- equations [23, 24]: tional efficiency of the differential quadrature method in this problem is demonstrated below in several case studies. The T T differential quadrature method is a relatively new method δΠ = δ[ε] [σ] − δ[w ] f dV = 0, (11) V that was introduced by Bellman and Casti [26]. After its appearance, several researchers have applied the differential quadrature method to solve a variety of problems in different where [ε] is a strain matrix, [σ] is a stress matrix, [w ]isa fields of science and engineering. The differential quadrature displacement matrix, and [ f ] is a inertia force matrix. The finite-element method can convert a differential equation method has been shown to be a powerful contender in solving initial and boundary value problems. Bert et al. into a set of algebraic equations, assume the shape of the solution in the element domain, and satisfy equilibrium. The [27–30] solved static and free vibration analysis of beams, Advances in Acoustics and Vibration 5 plates, and compressible lubrication using the differential After substituting (20)to(18)and (19), differential weight- quadrature method. Chen and Zhong [31] reported that, ing coefficients are given as follows: due to their global domain properties, differential quadrature (1) M (x ) and differential cubature methods could solve nonlinear pro- A = , im ( ) ( ) x − x M x blems more efficiently than traditional numerical techniques i m m such as the finite-element and the finite-difference methods. for i= m, i = 1, 2,... , N , m = 1, 2,... , N , / x x Civan and Sliepcevich [32] solved multivariable mathemat- ical models using the quadrature method and the cubature (1) (1) method. Han and Liew [33] analyzed the axisymmetric A = − A , ii im free vibration of moderately thick annular plates using the m=1 m= i differential quadrature methodology. Xu and Mazumder (27) [34] derived the rational ABCD matrix representing the L y (1) i B = , im high-speed interconnect using the differential quadrature ∗ y − y L (x ) i m m method. The differential quadrature method assumes that for i= m, i = 1, 2,... , N , m = 1, 2,... , N , / y y the derivative of a function at a sampling point can be Ny approximated as a weighted linear combination of the fun- (1) (1) B = − B . ctional values at all of the sampling points in the domain. ii im m=1 The number of equations is dependent upon the selected m= i number of the sampling points. A differential quadrature (1) approximation at the ith discrete point on a grid in the x-axis The weighting matrix A is [27–30] direction can be approximated by [29] ⎡ ⎤ (1) A [0] ··· [0] im ⎢ ⎥ ∂ W x , y (m) ⎢ ⎥ = A W x , y , ij ⎢ ⎥ (1) ∂x ⎢ . ⎥ j=1 (18) [0] A ··· . ⎢ ⎥ im (1) ⎢ ⎥ A = , ⎢ ⎥ . . ⎢ ⎥ for i = 1, 2,... , N . . . . (28) ⎢ . ⎥ . . [0] ⎢ ⎥ ⎣ ⎦ Adifferential quadrature approximation at the ith discrete (1) [0][0] ··· A im point on a grid in the direction of y-axis direction may be N N ×N N x y x y written as for i = 1, 2,... , N , m = 1, 2,... , N . x x ∂ W x, y (m) = B W x, y , ij (1) ∂y The weighting matrix B is [27–30] j=1 (19) ⎡ ⎤ (1) (1) (1) B [I] B [I] ··· B [I] for i = 1, 2,... , N , y 11 12 1N ⎢ ⎥ ⎢ ⎥ (1) (1) (1) (m) (m) ⎢ ⎥ B [I] B [I] ··· B [I] ⎢ ⎥ where A and B are the differential weighting coeffi- 21 22 2N ij ij ⎢ ⎥ (1) B = ⎢ ⎥ , (29) cients. The test function can be written as . . . ⎢ . ⎥ . . . . ⎢ ⎥ . . . ⎢ ⎥ W x, y = α(x)β y , (20) ⎣ ⎦ (1) (1) (1) B [I] B [I] ··· B [I] N 1 N 2 N N y y y y N N ×N N x y x y M(x) α(x) = ,for k = 1, 2,... , N , (21) ( ) ( ) x − x M x k k (1) where [I] is an identity matrix of dimension N .The A (1) x and B are both square matrices of dimension N N .The x y M(x) = (x − x ), (22) higher-order derivates may be obtained using the following m=1 equations: N N x y M (x ) = (x − x ),for k = 1, 2,... , N , (23) k k m x (m) (1) (m−1) A = A A , i,j i,k k,j m=1 m= / k k=1 N N x y L y (n) (1) (n−1) β(x) = ,for l = 1, 2,... , N , (24) (30) ∗ B = B B , y − y L y i,j i,k k,j l l k=1 Ny N N x y L y = y − y , (25) m (n) (m) (n) (m) A B = A B . i,j m=1 i,k k,j k=1 The above relation gives the higher order weighting coef- L y = y − y ,for l = 1, 2,... , N . l l m y (26) ficient matrix based on the first-order derivative weighting m=1 m= l coefficients. The selection of locations of the sampling points 6 Advances in Acoustics and Vibration is important for ensuring the accuracy of the solution of following nonuniform grid spacing gives better and more differential equations. Using equally spaced points can be reliable calculation results. The inner points are considered a convenient and easy selection method. The 1 (i − 1)π domain is divided by N × N points. The equally spaced x y x = 1 − cos ,for i = 1, 2,... , N , (33) i x sampling points are [29] 2 N − 1 i − 1 for x direction, x = ,for i = 1, 2,... , N , (31) i x N − 1 1 (i − 2)π y = 1 − cos ,for i = 1, 2,... , N , (34) for x direction, and i y 2 N − 3 for y direction. The use of zeros of shifted Legendre i − 1 y = ,for i = 1, 2,... , N , (32) i y polynomials have been known to give good results. Although N − 1 a convenient and commonly used choice of sampling of for y direction. An accurate solution can be obtained by a quadrature grid is equally spaced points, nonuniformly choosing a set of unequally spaced sampling points. A simple spaced points generally achieve better accuracy than a and effective choice is the roots of shifted Chebyshev and quadrature solution does. Equations (18)and (19)are Legendre points. Bert et al. [27–30] demonstrated that the substituted for (9)and (10) as follows: (4) (2) (4) (4) (2) (4) (4) (2) (4) (2) (2) (2) DA 2DA B DB DA 2DA B DB DA 2DA B DB i,N N i,N N i,N N x y x y x y i,1 i,1 i,1 i,2 i,2 i,2 + + + + ··· + + 4 2 2 4 4 2 2 4 4 2 2 4 a a b b a a b b a a b b W W ··· W × = ω ρhW ,for i = 1, 2, 3,... , N N − 1, N N , 1 2 N N i x y x y x y [W ] = [0],for i = 1, N , N − 1 N +1, N N , i x y x x y ⎡ ⎤ (2) (2) (2) (2) (2) (2) A vB A vB A vB T i,N N i,N N x y x y i,1 i,1 i,2 i,2 ⎣ ⎦ W W ··· W [ ] + + ··· + 1 2 N N = 0 , x y 2 2 2 2 2 2 a b a b a b for i = N +1,2N +1,3N +1,4N +1,... , N − 3 N +1, N − 2 N +1, x x x x y x y x 2N ,3N ,4N ,5N ,... , N − 2 N , N − 1 N , x x x x y x y x (3) (1) (3) (1) (3) (1) (2) (2) (2) A (2 − v)A B ( ) ( ) A 2 − v A B A 2 − v A B i,N N i,N N x y x y i,1 i,1 i,2 i,2 + + ··· + 3 2 3 2 3 2 a a b a a b a a b × W W ··· W = [0], 1 2 N N x y (35) for i = N +1,2N +1,3N +1,4N +1,... , N − 3 N +1, N − 2 N +1, x x x x y x y x 2N ,3N ,4N ,5N ,... , N − 2 N , N − 1 N , x x x x y x y x (2) (2) (2) (2) (2) (2) B vA B vA B vA i,NxNy i,NxNy i,1 i,1 i,2 i,2 W W ··· W 1 2 N N = [0], x y + + ··· + 2 2 2 2 2 2 b a b a b a for i = 2, 3, 4, 5,... , N − 2, N − 1, N − 1 N +2, N − 1 N +3, N − 1 N +4, x x y x y x y x N − 1 N +5,... , N N − 2, N N − 1, y x x y x y (3) (1) (3) (1) (3) (1) (2) (2) (2) ( ) B 2 − v B A B (2 − v)B A B (2 − v)B A i,N N i,N N i,1 i,1 i,2 i,2 x y x y + + ··· + 3 2 3 2 3 2 b b a b b a b b a × W W ··· W = [0], 1 2 N N x y for i = 2, 3, 4, 5,... , N − 2, N − 1, N − 1 N +2, N − 1 N +3, N − 1 N +4, x x y x y x y x N − 1 N +5,... , N N − 2, N N − 1. y x x y x y Advances in Acoustics and Vibration 7 1.4E +6 3E +6 1.2E +6 2.5E +6 1E +6 2E +6 8E +5 1.5E +6 6E +5 1E +6 4E +5 5E +5 2E +5 0E +0 0E +0 230 280 330 380 430 480 530 580 630 680 7 121722 27 3237 Length (micrometer) Thickness (micrometer) Mode 1 (differential quadrature method) Mode 1 (differential quadrature method) Mode 1 (finite-element method) Mode 1 (finite-element method) Mode 2 (differential quadrature method) Mode 2 (differential quadrature method) Mode 2 (finite-element method) Mode 2 (finite-element method) Mode 3 (differential quadrature method) Mode 3 (differential quadrature method) Mode 3 (finite-element method) Mode 3 (finite-element method) Figure 6: The lowest six frequencies of electrostatic actuator for Figure 8: The lowest six frequencies of electrostatic actuator for various lengths. various thicknesses. 1.4E +6 1.2E +6 1E +6 show that higher lengths produce smaller frequencies of the 8E +5 electrostatic actuator. The differential quadrature method 6E +5 has become a preferred method to the finite-element 4E +5 method. Figure 7 plots the frequencies of the electrostatic 2E +5 actuator with various widths. The length and thickness 0E +0 150 200 250 300 350 400 450 500 550 of the electrostatic actuator are 262 and 8 micrometers, Width (micrometer) respectively. The numerical results in this example show that the widths can significantly affect the dynamic behavior of Mode 1 (differential quadrature method) Mode 1 (finite-element method) the electrostatic actuator. Higher widths produce smaller Mode 2 (differential quadrature method) frequencies of the electrostatic actuator. Figure 8 shows the Mode 2 (finite-element method) frequencies of the microelectrostatic actuator with various Mode 3 (differential quadrature method) thicknesses. The length and width of the electrostatic actua- Mode 3 (finite-element method) tor are 262 and 176 micrometers, respectively. The numerical Figure 7: The lowest six frequencies of electrostatic actuator for results indicated that the frequency of the microactuator is various widths. increased for the actuator with a larger value of thickness. The numerical results indicate that the thickness of the actuator is a very sensitive parameter to the frequency of the actuator. The frequency of the actuator increases with Equation (35) is solved to obtain the frequencies of the increases in thickness h. electrostatic microactuators. 6. Frequencies of Electrostatic Micro Actuators 7. Conclusions The material parameters of the electrostatic actuator are ρ = 2.328 × 10 kg/m and E = 150 GPa [35]. The Numerical results indicate that length and width significantly FEMLAB finite-element package is used to analyze the impact the capacity of electrostatic microactuator. The pre- model. The 26733 solid elements are used to calculate the sented formulation reveals that the differential quadrature frequency of the actuator. Figure 6 plots the frequencies of approach is convenient for solving problems governed by the microelectrostatic actuator with various lengths. The fourth- or higher-order differential equations. Simulation width and thickness of the electrostatic actuator are 176 and results verify that the differential quadrature method obtains 8 micrometers, respectively. The numerical results indicate accurate results with relatively minimal computational and that the frequencies calculated using the finite-element modeling efforts. Length, width, and thickness can markedly method and the differential quadrature method are almost affect electrostatic microactuator frequencies. The FEMLAB identical. Several case studies have validated the applicability can handle capacitance and dynamic problems as well. of the method for solving such engineering problems. 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Dynamic Analysis of Electrostatic Microactuators Using the Differential Quadrature Method

Advances in Acoustics and Vibration , Volume 2011 – Aug 2, 2011

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Copyright © 2011 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.
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 576079, 8 pages doi:10.1155/2011/576079 Research Article Dynamic Analysis of Electrostatic Microactuators Using the Differential Quadrature Method Ming-Hung Hsu Department of Electrical Engineering, National Penghu University of Science and Technology, Penghu, Taiwan Correspondence should be addressed to Ming-Hung Hsu, hsu.mh@msa.hinet.net Received 3 January 2011; Accepted 26 April 2011 Academic Editor: Andrew Fleming Copyright © 2011 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. This work studies the dynamic behavior of electrostatic actuators using finite-element package software (FEMLAB) and differential quadrature method. The differential quadrature technique is used to transform partial differential equations into a discrete eigenvalue problem. Numerical results indicate that length, width, and thickness significantly impact the frequencies of the electrostatic actuators. The thickness could not affect markedly the electrostatic actuator capacities. The effects of varying actuator length, width, and thickness on the dynamic behavior and actuator capacities in electrostatic actuator systems are investigated. The differential quadrature method is an efficient differential equation solver. 1. Introduction model, the one-dimensional numerical model, and the finite-element model that incorporates a three-dimensional Plate-type electrostatic actuators are widely applied in simulation, were proposed to calculate the pull-in behaviors microelectromechanical systems. Microelectrostatic actua- of various fixed-fixed DMD structures and pressure sensors tor devices have a high operating frequency, low-power [8]. Gilbert et al. [9] analyzed the three-dimensional coupled consumption and can replace many passive components. electromechanics of MEMS using a CoSolve-EM simulation Mehdaoui et al. [1] presented the vertical cointegration of algorithm. Elwenspoek et al. [10] studied the dynamic AlSi MEMS tunable capacitors and Cu inductors for tunable behavior of active joints for various electrostatic actuator LC blocks. Etxeberria and Gracia [2] presented tunable designs. Shi et al. [11] presented a combination of an exterior MEMS volume capacitors for high-voltage applications. Liu boundary element method for electrostatics combined with et al. [3] presented that actuation by electrostatic repulsion a finite-element method for elasticity to evaluate the effect is produced by nonvolatile charge injection. Gallant and of coupling between the electrostatic force and the elastic Wood [4] investigated how fabrication techniques affect the deformation. Osterberg and Senturia [12] applied the sharp performance of widely tunable micromachined capacitors. instability phenomena of electrostatic pull-in behaviors for Borwick III et al. [5] analyzed a high Q, large tuning cantilever beam and fixed-fixed beam actuators to elicit range MEMS capacitor for RF filter systems. Harsh et al. the material characteristics of MEMS. Gretillat et al. [13] [6] studied the realization and design considerations of a employed three-dimensional MEMCAD and FEM programs flip-chip integrated MEMS tunable capacitor. Petersen [7] to simulate the dynamics of a nonlinear actuator, considering first described the nonlinear pull-in behavior of an electro- the effect of squeeze-film damping. Hung and Senturia [14] static microactuator. Osterberg et al. [8] proposed different developed leveraged bending and strain-stiffening methods numerical models for analyzing electrostatically deformed to increase the limiting travel distance before pull-in of diaphragms. Their results revealed that the electrostatic electrostatic actuators. Chan et al. [15] measured the pull-in deformation calculated using the one-dimensional model voltage and capacitance-voltage and performed 2D simula- is close to that obtained using a three-dimensional model. tions that included the electrical effects of fringing fields and Various models, including the lumped parallel-plate spring finite-beam thickness to determine the material properties of 2 Advances in Acoustics and Vibration 1.4E − 12 1.2E − 12 1E − 12 8E − 13 6E − 13 4E − 13 2E − 13 0E +0 02468 10 12 14 Electrode −5 ×10 176 micrometer 236 micrometer 196 micrometer 256 micrometer Figure 1: Schematic view of electrostatic actuator. 216 micrometer 276 micrometer Figure 3: Capacities of microelectrostatic actuator for various 1.2E − 12 widths. 1E − 12 8E − 13 6E − 13 1E − 12 9E − 13 4E − 13 8E − 13 2E − 13 7E − 13 0E +0 6E − 13 5E − 13 02468 10 12 14 4E − 13 −5 ×10 3E − 13 2E − 13 1E − 13 262 micrometer 322 micrometer 0E +0 282 micrometer 342 micrometer 02468 10 12 14 −5 302 micrometer 362 micrometer × 10 Figure 2: Capacities of microelectrostatic actuator for various 38 micrometer 98 micrometer lengths. 58 micrometer 118 micrometer 78 micrometer 138 micrometer Figure 4: Capacities of microelectrostatic actuator for various gap electrostatic microactuators. Li and Aluru [16] developed a distances. mixed-regime approach for combining linear and nonlinear theories to analyze large MEMS deformations at large applied voltages. Chyuan et al. [17–19] established the validity and partial differential equations and integral equations. The accuracy of the dual boundary element method and applied convergence and accuracy of the finite-element solution it to study the effect of gap size variation for the levitation of depends on the differential equation, integral form, and MEMS comb drive. Lai and Chen [20] studied the influence element used. The energy stored in the microdevice is of the holes in the membrane structures of radiofrequency expressed as follows [22–24]: MEMS switches. Qiao et al. [21] presented the suspension beam called two-beam to achieve parallel-plate actuator with W = ρ V dv,(1) e v extended working range, but without penalties of complex control circuit and large actuation voltage. In this work, where ρ is the volume charge density, v is the volume, and the finite-element method and the differential quadrature V is the applied voltage: method are employed to analyze how actuator length, width, and thickness affect dynamic behavior and capacitances in ρ = ∇· D,(2) v e electrostatic actuator systems. where D is the vector of electric flux density, 2. Capacitance of Electrostatic Micro Actuators 2W C =,(3) Figure 1 depicts the geometry of an electrostatic actuator where h is the thickness of the movable plate [22]. The length where C is the capacitance. The capacitance models are and width of the movable plate are a and b,respectively. extracted using three-dimensional simulations. Figure 2 The microactuator design is based upon the deformation of shows the capacitances of the electrostatic microactuator a movable mechanical structure by electrostatic forces using with various lengths. The width, thickness, and gap distance a fixed electrode. A voltage applied across the gap creates of the electrostatic actuator are 176, 8, and 38 micrometers, the electrostatic force. Electrostatic actuator devices are fabri- respectively. The computational solution is acquired by the cated from polysilicon by surface micromaching techniques. finite-element scheme. The FEMLAB finite-element package In this work, the finite-element technique is applied to find is used to analyze the model. In previous cases, 62463 solid the capacitance of the electrostatic microactuator. The finite- element method is used to find approximate solutions for elements are used to calculate the capacitance of the actuator Farad C (Farad) C (Farad) Advances in Acoustics and Vibration 3 9.08E − 14 where δW is the virtual work. This leads to the following 9.07E − 14 equations for electrostatic actuator motion: 9.06E − 14 9.05E − 14 9.04E − 14 4 4 ∂ w x, y, t ∂ w x, y, t 9.03E − 14 D +2D 9.02E − 14 4 2 2 ∂x ∂x ∂y 9.01E − 14 (7) 0 2 4 6 8 10 12 14 16 18 4 2 ∂ w x, y, t ∂ w x, y, t −5 ×10 V + D + ρh = 0, 4 2 ∂y ∂t 8 micrometer 14 micrometer 16 micrometer 10 micrometer 12 micrometer The boundary conditions of the actuator are as follows: Figure 5: Capacities of microelectrostatic actuator for various thicknesses. w(0, 0, t) = 0, w(a,0, t) = 0, and provide convergent results. The effect of the actuator ( ) w 0, b, t = 0, length is more pronounced at higher capacity. Figure 3 shows the capacitances of the electrostatic microactuator for w(a, b, t) = 0, varying widths. The length, thickness, and gap distance of 2 2 ∂ w x, y, t ∂ w x, y, t the electrostatic actuator are 262, 8, and 38 micrometers, + ν = 0, 2 2 respectively. Numerical results reveal that the capacity of the ∂x ∂y actuator increases as the width of the actuator increases. for x = 0, 0 <y < b, Figure 4 shows the capacitances of electrostatic microac- 3 3 ∂ w x, y, t ∂ w x, y, t tuator for various gap distances. The width, length, and + (2 − ν) = 0, 3 2 ∂x ∂x ∂y thickness of the electrostatic actuator are 176, 262, and for x = 0, 0 <y < b, 8 micrometers, respectively. Numerical results reveal that the capacity of the actuator increases as the gap distance 2 2 ∂ w x, y, t ∂ w x, y, t of the actuator falls. Figure 5 shows the capacitances of + ν = 0, 2 2 the electrostatic microactuator at various thicknesses. The ∂x ∂y width, length, and gap distance of the electrostatic actuator for x = a,0 <y < b, are 176, 262, and 38 micrometers, respectively. Numerical (8) 3 3 results show that the thickness does not significantly affect ∂ w x, y, t ∂ w x, y, t + (2 − ν) = 0, 3 2 the capacitances of the electrostatic microactuator. The ∂x ∂x ∂y evaluation illustrates capacitance problems. for x = a,0 <y < b, 2 2 ∂ w x, y, t ∂ w x, y, t 3. Vibration Analysis of Electrostatic + ν = 0, 2 2 ∂y ∂x Micro Actuators for 0 <x < a, y = 0, The electrostatic actuator has length a, width b, and thickness 3 3 ∂ w x, y, t ∂ w x, y, t h. The strain energy of the actuator is [25] + (2 − ν) = 0, 3 2 ∂y ∂x ∂y for 0 <x < a, y = 0, b a 2 2 1 ∂ w x, y, t ∂ w x, y, t U = D + dx dy,(4) 2 2 2 2 2 ∂x ∂y ∂ w x, y, t ∂ w x, y, t 0 0 + ν = 0, 2 2 ∂y ∂x where w is the deflection of the actuator, D = Eh /(12(1 − for 0 <x < a, y = b, ν )) is the flexural rigidity, E is Young’s modulus, ν is the 3 3 Poisson’s ratio, and h is the actuator thickness. The kinetic ∂ w x, y, t ∂ w x, y, t ( ) + 2 − ν = 0, 3 2 energy of the microactuator is ∂y ∂x ∂y for 0 <x < a, y = b, b a 1 ∂w x, y, t T = ρh dx dy,(5) 2 ∂t 0 0 iωt After substituting w(x, y, t) = W(x, y)e into (7), (7)can be rewritten as follows: where t is the time and ρ is the density of the actuator material. Equations (4)and (5) are substituted into Hamilton 4 4 4 equation as follows: ∂ W x, y ∂ W x, y ∂ W x, y D +2D + D 4 2 2 4 ∂x ∂x ∂y ∂y (9) (δT − δU + δW)dt = 0, (6) = ω ρhW x, y , C (Farad) 4 Advances in Acoustics and Vibration where ω is the natural frequency of the actuator. The stress-strain relationship for linear conditions is as follows: boundary conditions of the electrostatic actuator are as follows: [σ] = D [ε], (12) where [D] is an elastic matrix. The variables are approxi- ( ) W 0, 0 = 0, mated with functions in the chosen finite-element spaces. W(a,0) = 0, The following finite-element solution is assumed [23, 24]: W(0, b) = 0, ∗ ∗ w = N [W ] cos(ωt), (13) W(a, b) = 0, where [N] is the matrix of any suitable assumed shape 2 2 function and [W ] is the displacement matrix. The finite ∂ W x, y ∂ W x, y + ν = 0, element assembles all elements to form a complete structure 2 2 ∂x ∂x to equilibrate a structure with its environment. The equation for x = 0, 0 <y < b, for the finite-element model of the electrostatic microactua- 3 3 ∂ W x, y ∂ W x, y tor is as follows: + (2 − ν) = 0, 3 2 ∂x ∂x ∂y ∗ ∗ 2 ∗ ∗ [K ][W ] = ω [M ][W ], (14) for x = 0, 0 <y < b, ∗ ∗ 2 2 where [M ] is the mass matrix and [K ] is the stiffness ∂ W x, y ∂ W x, y + ν = 0, matrix. The stiffness matrix can be written as follows: 2 2 ∂x ∂y for x = a,0 <y < b, T [K ] = L N D L N dV , (15) (10) 3 3 ∂ W x, y ∂ W x, y + (2 − ν) = 0, 3 2 ∂x ∂x ∂y where [L] is a linear differential operator matrix. The mass for x = a,0 <y < b, matrix can be written as follows: 2 2 T ∂ W x, y ∂ W x, y ∗ [M ] = N ρ N dV. (16) + ν = 0, 2 2 ∂y ∂x for 0 <x < a, y = 0, ∗ ∗ ∗ The matrix order is N × N ,where N is the number 3 3 of nodes for which the solution is unknown. Assembly of ∂ W x, y ∂ W x, y + (2 − ν) = 0, all element stiffness matrices and element mass matrices of 3 2 ∂y ∂x ∂y the electrostatic actuator defines the following eigenvalue for 0 <x < a, y = 0, equation: 2 2 ∂ W x, y ∂ W x, y ∗ 2 ∗ + ν = 0, [K ] − ω [M ] = 0. (17) 2 2 ∂y ∂x for 0 <x < a, y = b, The eigenvalues of the electrostatic actuator can be derived from (17) and are known as eigenfrequency problems. 3 3 ∂ W x, y ∂ W x, y ( ) + 2 − ν = 0, 3 2 ∂y ∂x ∂y 5. Differential Quadrature Formula for 0 <x < a, y = b, The vibration response of the microactuator is numerically 4. Finite-element Method Model modeled using the differential quadrature method in this work. The differential quadrature method is used to convert The commercially available FEMLAB software package is the partial differential equations of the plates into a discrete used to evaluate dynamic problems based on partial differ- eigenvalue problem. The roots of shifted Chebyshev and ential equations. To derive finite formulations, the following Legendre sampling point equation are used to select the sam- virtual work principle must be utilized in the following pling points in these analyses. The integrity and computa- equations [23, 24]: tional efficiency of the differential quadrature method in this problem is demonstrated below in several case studies. The T T differential quadrature method is a relatively new method δΠ = δ[ε] [σ] − δ[w ] f dV = 0, (11) V that was introduced by Bellman and Casti [26]. After its appearance, several researchers have applied the differential quadrature method to solve a variety of problems in different where [ε] is a strain matrix, [σ] is a stress matrix, [w ]isa fields of science and engineering. The differential quadrature displacement matrix, and [ f ] is a inertia force matrix. The finite-element method can convert a differential equation method has been shown to be a powerful contender in solving initial and boundary value problems. Bert et al. into a set of algebraic equations, assume the shape of the solution in the element domain, and satisfy equilibrium. The [27–30] solved static and free vibration analysis of beams, Advances in Acoustics and Vibration 5 plates, and compressible lubrication using the differential After substituting (20)to(18)and (19), differential weight- quadrature method. Chen and Zhong [31] reported that, ing coefficients are given as follows: due to their global domain properties, differential quadrature (1) M (x ) and differential cubature methods could solve nonlinear pro- A = , im ( ) ( ) x − x M x blems more efficiently than traditional numerical techniques i m m such as the finite-element and the finite-difference methods. for i= m, i = 1, 2,... , N , m = 1, 2,... , N , / x x Civan and Sliepcevich [32] solved multivariable mathemat- ical models using the quadrature method and the cubature (1) (1) method. Han and Liew [33] analyzed the axisymmetric A = − A , ii im free vibration of moderately thick annular plates using the m=1 m= i differential quadrature methodology. Xu and Mazumder (27) [34] derived the rational ABCD matrix representing the L y (1) i B = , im high-speed interconnect using the differential quadrature ∗ y − y L (x ) i m m method. The differential quadrature method assumes that for i= m, i = 1, 2,... , N , m = 1, 2,... , N , / y y the derivative of a function at a sampling point can be Ny approximated as a weighted linear combination of the fun- (1) (1) B = − B . ctional values at all of the sampling points in the domain. ii im m=1 The number of equations is dependent upon the selected m= i number of the sampling points. A differential quadrature (1) approximation at the ith discrete point on a grid in the x-axis The weighting matrix A is [27–30] direction can be approximated by [29] ⎡ ⎤ (1) A [0] ··· [0] im ⎢ ⎥ ∂ W x , y (m) ⎢ ⎥ = A W x , y , ij ⎢ ⎥ (1) ∂x ⎢ . ⎥ j=1 (18) [0] A ··· . ⎢ ⎥ im (1) ⎢ ⎥ A = , ⎢ ⎥ . . ⎢ ⎥ for i = 1, 2,... , N . . . . (28) ⎢ . ⎥ . . [0] ⎢ ⎥ ⎣ ⎦ Adifferential quadrature approximation at the ith discrete (1) [0][0] ··· A im point on a grid in the direction of y-axis direction may be N N ×N N x y x y written as for i = 1, 2,... , N , m = 1, 2,... , N . x x ∂ W x, y (m) = B W x, y , ij (1) ∂y The weighting matrix B is [27–30] j=1 (19) ⎡ ⎤ (1) (1) (1) B [I] B [I] ··· B [I] for i = 1, 2,... , N , y 11 12 1N ⎢ ⎥ ⎢ ⎥ (1) (1) (1) (m) (m) ⎢ ⎥ B [I] B [I] ··· B [I] ⎢ ⎥ where A and B are the differential weighting coeffi- 21 22 2N ij ij ⎢ ⎥ (1) B = ⎢ ⎥ , (29) cients. The test function can be written as . . . ⎢ . ⎥ . . . . ⎢ ⎥ . . . ⎢ ⎥ W x, y = α(x)β y , (20) ⎣ ⎦ (1) (1) (1) B [I] B [I] ··· B [I] N 1 N 2 N N y y y y N N ×N N x y x y M(x) α(x) = ,for k = 1, 2,... , N , (21) ( ) ( ) x − x M x k k (1) where [I] is an identity matrix of dimension N .The A (1) x and B are both square matrices of dimension N N .The x y M(x) = (x − x ), (22) higher-order derivates may be obtained using the following m=1 equations: N N x y M (x ) = (x − x ),for k = 1, 2,... , N , (23) k k m x (m) (1) (m−1) A = A A , i,j i,k k,j m=1 m= / k k=1 N N x y L y (n) (1) (n−1) β(x) = ,for l = 1, 2,... , N , (24) (30) ∗ B = B B , y − y L y i,j i,k k,j l l k=1 Ny N N x y L y = y − y , (25) m (n) (m) (n) (m) A B = A B . i,j m=1 i,k k,j k=1 The above relation gives the higher order weighting coef- L y = y − y ,for l = 1, 2,... , N . l l m y (26) ficient matrix based on the first-order derivative weighting m=1 m= l coefficients. The selection of locations of the sampling points 6 Advances in Acoustics and Vibration is important for ensuring the accuracy of the solution of following nonuniform grid spacing gives better and more differential equations. Using equally spaced points can be reliable calculation results. The inner points are considered a convenient and easy selection method. The 1 (i − 1)π domain is divided by N × N points. The equally spaced x y x = 1 − cos ,for i = 1, 2,... , N , (33) i x sampling points are [29] 2 N − 1 i − 1 for x direction, x = ,for i = 1, 2,... , N , (31) i x N − 1 1 (i − 2)π y = 1 − cos ,for i = 1, 2,... , N , (34) for x direction, and i y 2 N − 3 for y direction. The use of zeros of shifted Legendre i − 1 y = ,for i = 1, 2,... , N , (32) i y polynomials have been known to give good results. Although N − 1 a convenient and commonly used choice of sampling of for y direction. An accurate solution can be obtained by a quadrature grid is equally spaced points, nonuniformly choosing a set of unequally spaced sampling points. A simple spaced points generally achieve better accuracy than a and effective choice is the roots of shifted Chebyshev and quadrature solution does. Equations (18)and (19)are Legendre points. Bert et al. [27–30] demonstrated that the substituted for (9)and (10) as follows: (4) (2) (4) (4) (2) (4) (4) (2) (4) (2) (2) (2) DA 2DA B DB DA 2DA B DB DA 2DA B DB i,N N i,N N i,N N x y x y x y i,1 i,1 i,1 i,2 i,2 i,2 + + + + ··· + + 4 2 2 4 4 2 2 4 4 2 2 4 a a b b a a b b a a b b W W ··· W × = ω ρhW ,for i = 1, 2, 3,... , N N − 1, N N , 1 2 N N i x y x y x y [W ] = [0],for i = 1, N , N − 1 N +1, N N , i x y x x y ⎡ ⎤ (2) (2) (2) (2) (2) (2) A vB A vB A vB T i,N N i,N N x y x y i,1 i,1 i,2 i,2 ⎣ ⎦ W W ··· W [ ] + + ··· + 1 2 N N = 0 , x y 2 2 2 2 2 2 a b a b a b for i = N +1,2N +1,3N +1,4N +1,... , N − 3 N +1, N − 2 N +1, x x x x y x y x 2N ,3N ,4N ,5N ,... , N − 2 N , N − 1 N , x x x x y x y x (3) (1) (3) (1) (3) (1) (2) (2) (2) A (2 − v)A B ( ) ( ) A 2 − v A B A 2 − v A B i,N N i,N N x y x y i,1 i,1 i,2 i,2 + + ··· + 3 2 3 2 3 2 a a b a a b a a b × W W ··· W = [0], 1 2 N N x y (35) for i = N +1,2N +1,3N +1,4N +1,... , N − 3 N +1, N − 2 N +1, x x x x y x y x 2N ,3N ,4N ,5N ,... , N − 2 N , N − 1 N , x x x x y x y x (2) (2) (2) (2) (2) (2) B vA B vA B vA i,NxNy i,NxNy i,1 i,1 i,2 i,2 W W ··· W 1 2 N N = [0], x y + + ··· + 2 2 2 2 2 2 b a b a b a for i = 2, 3, 4, 5,... , N − 2, N − 1, N − 1 N +2, N − 1 N +3, N − 1 N +4, x x y x y x y x N − 1 N +5,... , N N − 2, N N − 1, y x x y x y (3) (1) (3) (1) (3) (1) (2) (2) (2) ( ) B 2 − v B A B (2 − v)B A B (2 − v)B A i,N N i,N N i,1 i,1 i,2 i,2 x y x y + + ··· + 3 2 3 2 3 2 b b a b b a b b a × W W ··· W = [0], 1 2 N N x y for i = 2, 3, 4, 5,... , N − 2, N − 1, N − 1 N +2, N − 1 N +3, N − 1 N +4, x x y x y x y x N − 1 N +5,... , N N − 2, N N − 1. y x x y x y Advances in Acoustics and Vibration 7 1.4E +6 3E +6 1.2E +6 2.5E +6 1E +6 2E +6 8E +5 1.5E +6 6E +5 1E +6 4E +5 5E +5 2E +5 0E +0 0E +0 230 280 330 380 430 480 530 580 630 680 7 121722 27 3237 Length (micrometer) Thickness (micrometer) Mode 1 (differential quadrature method) Mode 1 (differential quadrature method) Mode 1 (finite-element method) Mode 1 (finite-element method) Mode 2 (differential quadrature method) Mode 2 (differential quadrature method) Mode 2 (finite-element method) Mode 2 (finite-element method) Mode 3 (differential quadrature method) Mode 3 (differential quadrature method) Mode 3 (finite-element method) Mode 3 (finite-element method) Figure 6: The lowest six frequencies of electrostatic actuator for Figure 8: The lowest six frequencies of electrostatic actuator for various lengths. various thicknesses. 1.4E +6 1.2E +6 1E +6 show that higher lengths produce smaller frequencies of the 8E +5 electrostatic actuator. The differential quadrature method 6E +5 has become a preferred method to the finite-element 4E +5 method. Figure 7 plots the frequencies of the electrostatic 2E +5 actuator with various widths. The length and thickness 0E +0 150 200 250 300 350 400 450 500 550 of the electrostatic actuator are 262 and 8 micrometers, Width (micrometer) respectively. The numerical results in this example show that the widths can significantly affect the dynamic behavior of Mode 1 (differential quadrature method) Mode 1 (finite-element method) the electrostatic actuator. Higher widths produce smaller Mode 2 (differential quadrature method) frequencies of the electrostatic actuator. Figure 8 shows the Mode 2 (finite-element method) frequencies of the microelectrostatic actuator with various Mode 3 (differential quadrature method) thicknesses. The length and width of the electrostatic actua- Mode 3 (finite-element method) tor are 262 and 176 micrometers, respectively. The numerical Figure 7: The lowest six frequencies of electrostatic actuator for results indicated that the frequency of the microactuator is various widths. increased for the actuator with a larger value of thickness. The numerical results indicate that the thickness of the actuator is a very sensitive parameter to the frequency of the actuator. The frequency of the actuator increases with Equation (35) is solved to obtain the frequencies of the increases in thickness h. electrostatic microactuators. 6. Frequencies of Electrostatic Micro Actuators 7. Conclusions The material parameters of the electrostatic actuator are ρ = 2.328 × 10 kg/m and E = 150 GPa [35]. The Numerical results indicate that length and width significantly FEMLAB finite-element package is used to analyze the impact the capacity of electrostatic microactuator. The pre- model. The 26733 solid elements are used to calculate the sented formulation reveals that the differential quadrature frequency of the actuator. Figure 6 plots the frequencies of approach is convenient for solving problems governed by the microelectrostatic actuator with various lengths. The fourth- or higher-order differential equations. Simulation width and thickness of the electrostatic actuator are 176 and results verify that the differential quadrature method obtains 8 micrometers, respectively. The numerical results indicate accurate results with relatively minimal computational and that the frequencies calculated using the finite-element modeling efforts. Length, width, and thickness can markedly method and the differential quadrature method are almost affect electrostatic microactuator frequencies. The FEMLAB identical. Several case studies have validated the applicability can handle capacitance and dynamic problems as well. of the method for solving such engineering problems. 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