Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 586718, 25 pages
doi:10.1155/2010/586718
Research Article

A Wavelet Interpolation Galerkin Method for the Simulation of MEMS Devices under the Effect of Squeeze Film Damping

Pu Li1 and Yuming Fang2

1School of Mechanical Engineering, Southeast University, Jiangning, Nanjing 211189, China
2College of Electronic Science and Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China

Received 29 March 2009; Revised 17 September 2009; Accepted 27 October 2009

Academic Editor: Stefano Lenci

Copyright © 2010 Pu Li and Yuming Fang. 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.

Abstract

This paper presents a new wavelet interpolation Galerkin method for the numerical simulation of MEMS devices under the effect of squeeze film damping. Both trial and weight functions are a class of interpolating functions generated by autocorrelation of the usual compactly supported Daubechies scaling functions. To the best of our knowledge, this is the first time that wavelets have been used as basis functions for solving the PDEs of MEMS devices. As opposed to the previous wavelet-based methods that are all limited in one energy domain, the MEMS devices in the paper involve two coupled energy domains. Two typical electrically actuated micro devices with squeeze film damping effect are examined respectively to illustrate the new wavelet interpolation Galerkin method. Simulation results show that the results of the wavelet interpolation Galerkin method match the experimental data better than that of the finite difference method by about 10%.

1. Introduction

Modeling and simulation of MEMS devices play an important role in the design phase for system optimization and for the reduction of design cycles. The performances of MEMS devices are represented by partial-differential equations (PDEs) and associated boundary conditions. In the past two decades, there have been extensive, and successful, works focused on solving the partial-differential equations of MEMS [1–15]. A detailed review of the works is available in [1]. In the previous works, Galerkin method was widely used to reduce the partial-differential equations to ordinary-differential equations (ODEs) in time and then solve the reduced equations either numerically or analytically. The previous works differ from each other in the choice of the basis functions.

The basis set can be chosen arbitrarily, as long as its elements satisfy all of the boundary conditions and are sufficiently differentiable. To enhance convergence, the basis set has to be chosen to resemble the behavior of the device. For example, two ways have been used to generate the basis set for the reduced-order models of MEMS devices [1]. The first way [4, 9] uses the undamped linear model shapes of the undeflected microstructure as basis functions. For simple structures with simple boundary conditions, the mode shapes are found analytically. For complex structures or complex boundary conditions, the linear mode shapes are obtained numerically using the finite element method. The second way [2] conducts experiments or solves the PDEs using FEM or FDM to generate snapshots under a training signal, then applies a modal analysis method (one of the variation of the proper orthogonal decomposition method [6]) to the time series to extract the mode shapes of the device structural elements.

In the past two decades also, a new numerical concept was introduced and is gaining increasing popularity [16–25]. The method is based on the expansion of functions in terms of a set of basis functions called wavelets. Indeed wavelets have many excellent properties such as orthogonality, compact support, exact representation of polynomials to a certain degree, and flexibility to represent functions at different levels of resolution. Indeed a complete basis can be generated easily by a signal function through dilatation and translation. The wavelet-based methods may be classified as wavelet-Galerkin method [19, 20], wavelet-collocation method [21, 22], and wavelet interpolation Galerkin method [23–25]. Among the three methods, the wavelet-Galerkin method is the most common one because of its implementation simplicity. The method is a Galerkin scheme using scaling or wavelet functions as the trial and weight functions. However, both scaling and wavelet functions do not satisfy the boundary conditions. Thus the treatment of general boundary conditions is a major difficulty for the application of the wavelet-Galerkin method, especially for the bounded region problems, even though different efforts [19, 20] have been made. For the wavelet-collocation method, boundary conditions can be treated in a satisfactory way [21]. In the method, trial functions are a class of interpolating functions generated by autocorrelation of the usual compactly supported Daubechies scaling functions. However, the method requires the calculation of higher-order derivatives (up to the second derivatives for second-order parabolic problems) of the wavelets. Due to the derivatives of compactly supported wavelets being highly oscillatory, it is difficult to compute the connection coefficients by the numerical evaluation of integral [18]. The wavelet interpolation Galerkin method is a Galerkin scheme that both trial and weight functions are a class of interpolating functions generated by autocorrelation of the usual compactly supported Daubechies scaling functions. For the method, the boundary conditions [24] can be treated easily and the formulations are derived from the weak form; thus only the first derivatives of wavelets (for second-order parabolic problems) are required.

Wavelets have proven to be an efficient tool of analysis in many fields including the solution of PDEs. However, few papers in MEMS area give attention to the wavelet-based methods. This paper presents a new wavelet interpolation Galerkin method for the numerical simulation of MEMS devices under the effect of squeeze film damping. To the best of our knowledge, this is the first time that wavelets have been used as basis functions for solving the PDEs of MEMS devices. As opposed to the previous wavelet-based methods that are all limited in one energy domain, the MEMS devices in the paper involve two coupled energy domains. The squeeze film damping effect on the dynamics of microstructures has already been extensively studied. We stress that our intention here is not to discover new physics to the squeeze film damping.

The outline of this paper is as follows. Section 2 presents a brief introduction to some major concepts and properties of wavelets. In Sections 3 and 4, two typical electrically actuated micro devices with squeeze film damping effect are examined respectively to illustrate the wavelet interpolation Galerkin method. Section 5 calculates the frequency responses and the quality factors using the present method, and compares the calculated results with those generated by experiment [26, 27], by the finite difference method, and by other published analytical models [15, 26]. Finally, a conclusion is given in Section 6.

2. Basic Concepts of Daubechies’ Wavelets and Wavelet Interpolation

In this section, we shall give a brief introduction to the concepts and properties of Daubechies’ wavelets. More detailed discussions can be found in [16–18, 21].

2.1. Daubechies’ Orthonormal Wavelets

Daubechies [16, 17] constructed a family of orthonomal bases of compactly supported wavelets for the space of square-integrable funcntions, 𝐿 2 ( 𝑅 ) . Due to the fact that they possess several useful properties, such as orthogonality, compact support, exact representation of polynomials to a certain degree, and ability to represent functions at different levels of resolution, Daubechies’ wavelets have gained great interest in the numerical solutions of PDEs [18–22].

Daubechies’ functions are easy to construct [16, 17]. For an even integer 𝐿 , we have the Daubechies’ scaling function 𝜙 ( 𝑥 ) and wavelet 𝜓 ( 𝑥 ) satisfying 𝜙 ( 𝑥 ) = 𝐿 − 1 𝑖 = 0 ̂ 𝑝 𝑖 𝜓 𝜙 ( 2 𝑥 − 𝑖 ) ( 𝑥 ) = 1 𝑖 = 2 − 𝐿 ( − 1 ) 𝑖 ̂ 𝑝 1 − 𝑖 𝜙 ( 2 𝑥 − 𝑖 ) . ( 2 . 1 ) The fundamental support of the scaling function 𝜙 ( 𝑥 ) is in the interval [ 0 , 𝐿 − 1 ] while that of the corresponding wavelet 𝜓 ( 𝑥 ) is in the interval [ 1 − 𝐿 / 2 , 𝐿 / 2 ] . The parameter 𝐿 will be referred to as the degree of the scaling function 𝜙 ( 𝑥 ) . The coefficients ̂ 𝑝 𝑖 are called the wavelet filter coefficients. Daubenchies [16, 17] established these wavelet filter coefficients to satisfy the following conditions: 𝐿 − 1 𝑖 = 0 ̂ 𝑝 𝑖 = 2 , 𝐿 − 1 𝑖 = 0 ̂ 𝑝 𝑖 ̂ 𝑝 𝑖 − 𝑚 = 𝛿 0 , 𝑚 , 1 𝑖 = 2 − 𝐿 ( − 1 ) 𝑖 ̂ 𝑝 1 − 𝑖 ̂ 𝑝 𝑖 − 2 𝑚 = 0 f o r i n t e g e r , 𝐿 − 1 𝑖 = 0 ( − 1 ) 𝑖 𝑖 𝑚 ̂ 𝑝 𝑖 𝐿 = 0 , 𝑚 = 0 , 1 , … , 2 − 1 , ( 2 . 2 ) where 𝛿 0 , 𝑚 is the Kronecker delta function. Correspondingly, the constructed scanling function 𝜙 ( 𝑥 ) and wavelet 𝜓 ( 𝑥 ) have the following properties: ∞ − ∞ 𝜙 ( 𝑥 ) d 𝑥 = 1 , ∞ − ∞ 𝜙 ( 𝑥 − 𝑖 ) 𝜙 ( 𝑥 − 𝑚 ) d 𝑥 = 𝛿 𝑖 , 𝑚 , f o r i n t e g e r s 𝑖 , 𝑚 ∞ − ∞ 𝜙 ( 𝑥 ) 𝜓 ( 𝑥 − 𝑚 ) d 𝑥 = 0 f o r i n t e g e r 𝑚 , ∞ − ∞ 𝑥 𝑚 𝐿 𝜓 ( 𝑥 ) d 𝑥 = 0 , 𝑚 = 0 , 1 , … , 2 − 1 . ( 2 . 3 )

Denote by 𝐿 2 ( 𝑅 ) the space of square-integrable functions on the real line. Let 𝑉 𝐽 and 𝑊 𝐽 be the subspace generated, respectively, as the 𝐿 2 -closure of the linear spans of 𝜙 𝐽 , 𝑖 ( 𝑥 ) = 2 𝐽 / 2 𝜙 ( 2 𝐽 𝑥 − 𝑖 ) and 𝜓 𝐽 , 𝑖 ( 𝑥 ) = 2 𝐽 / 2 𝜓 ( 2 𝐽 𝑥 − 𝑖 ) , 𝐽 , 𝑖 ∈ 𝑍 . ℤ denotes the set of integers. Then (2.3) implies that 𝑉 𝐽 + 1 = 𝑉 𝐽 ⊕ 𝑊 𝐽 , 𝑉 0 ⊂ 𝑉 1 ⊂ ⋯ 𝑉 𝐽 ⊂ 𝑉 𝐽 + 1 , 𝑉 𝐽 + 1 = 𝑉 0 ⊕ 𝑊 0 ⊕ 𝑊 1 ⊕ ⋯ 𝑊 𝐽 , ( 2 . 4 ) Equation (2.4) presents the multiresolution properties of wavlets. Any function 𝑓 ∈ 𝐿 2 ( 𝑅 ) , may be approximated by the multiresolution apparatus described above, by its projection 𝑃 𝑉 𝐽 𝑓 onto the subspace 𝑉 𝐽 𝑃 𝑉 𝐽 𝑓 = 𝑖 ∈ 𝑍 𝑓 𝐽 , 𝑖 𝜙 𝐽 , 𝑖 ( 𝑥 ) . ( 2 . 5 )

2.2. Wavelet Interpolation Scaling Function

For a given Daubechies’ scaling function, its autocorrelation function 𝜃 ( 𝑥 ) can be defined as follows [21]: 𝜃 ( 𝑥 ) = ∞ − ∞ 𝜙 ( 𝜏 ) 𝜙 ( 𝜏 − 𝑥 ) d 𝜏 . ( 2 . 6 ) The function satisfies the following interpolating property 𝜃 ( 𝑘 ) = 𝛿 0 , 𝑘 , 𝑘 ∈ 𝑍 , ( 2 . 7 ) and has a symmetric support [ − ( 𝐿 − 1 ) , ( 𝐿 − 1 ) ] . The derivative of the function 𝜃 ( 𝑘 ) may be computed by differentiating the convolution product 𝜃 ( 𝑠 ) ( 𝑘 ) = ( − 1 ) ( 𝑠 ) ∞ − ∞ 𝜙 ( 𝜏 ) 𝜙 ( 𝑠 ) ( 𝜏 − 𝑘 ) d 𝜏 . ( 2 . 8 ) Let 𝜃 ( 𝑥 ) act as the scaling function, we have 𝜃 𝐽 , 𝑘 2 ( 𝑥 ) = 𝜃 𝐽 𝑥 − 𝑘 , 𝑘 ∈ 𝑍 . ( 2 . 9 ) For a set of dyadic grids of the type 𝑥 𝐽 𝑘 ∈ 𝑅 : 𝑥 𝐽 𝑘 = 2 − 𝐽 𝑘 , where 𝑘 , 𝐽 ∈ 𝑍 , the 𝜃 𝐽 , 𝑘 ( 𝑥 ) verifies the interpolation property at the dyadic points: 𝜃 𝐽 , 𝑘 ( 𝑥 𝐽 𝑛 ) = 𝛿 𝑛 , 𝑘 . Let 𝑉 𝑥 𝐽 be the linear span of the set { 𝜃 ( 2 𝐽 𝑥 − 𝑘 ) , 𝑘 ∈ 𝑍 } . It can be proved that { 𝑉 𝑥 𝐽 } forms a multiresolution analysis, where 𝜃 𝐽 , 𝑘 ( 𝑥 ) acts as the role of scaling function (the so-called interpolation scaling function), and the set { 𝜃 ( 2 𝐽 𝑥 − 𝑘 ) , 𝑘 ∈ 𝑍 } is a Riesz’s basis for 𝑉 𝑥 𝐽 . For a function 𝑓 ∈ 𝐻 1 ( 𝑅 ) , an interpolation operator 𝐼 𝐽 ∶ 𝐻 1 ( 𝑅 ) → 𝑉 𝑥 𝐽 can be defined [21]: 𝐼 𝐽 ( 𝑓 ) = 𝑘 𝑓 𝐽 𝑘 𝜃 2 𝐽 𝑥 − 𝑘 , 𝑘 ∈ 𝑍 , ( 2 . 1 0 ) where 𝑓 𝐽 𝑘 = 𝑓 ( 𝑥 𝐽 𝑘 ) = 𝑓 ( 2 − 𝐽 𝑘 ) . Thus, for a function 𝑓 ( 𝑥 ) defined on 𝑥 ∈ [ 0 , 1 ] , 𝑓 ( 𝑥 ) has the following approximation 𝑓 ( 𝑥 ) = 2 𝐽 + ( 𝐿 − 1 ) 𝑘 = − ( 𝐿 − 1 ) 𝑓 𝐽 , 𝑘 𝜃 2 𝐽 = 𝑥 − 𝑘 1 𝑘 = − ( 𝐿 − 1 ) 𝑓 𝐽 , 𝑘 𝜃 2 𝐽 + 𝑥 − 𝑘 2 𝐽 𝑘 = 0 𝑓 𝐽 , 𝑘 𝜃 2 𝐽 + 𝑥 − 𝑘 2 𝐽 + 𝐿 − 1 𝑘 = 2 𝐽 + 1 𝑓 𝐽 , 𝑘 𝜃 2 𝐽 . 𝑥 − 𝑘 ( 2 . 1 1 )

In this paper, wavelet collocation scheme is applied on 𝑥 ∈ [ 0 , 1 ] , where 𝑥 𝐽 𝑘 = 2 − 𝐽 𝑘 and 𝑘 = 0 , 1 , … , 2 𝐽 . Therefore, instead of the values of 𝑓 ( 𝑥 ) at 𝑥 𝐽 𝑘 , 𝑘 = − ( 𝐿 − 1 ) , … , − 1 and 𝑘 = 2 𝐽 + 1 , … , ( 𝐿 − 1 ) , we may use some values which are extrapolated from the values in those dyadic points internal to the interval 𝑥 ∈ [ 0 , 1 ] . As described in [21, 22], we define 𝑓 ( 𝑥 ) = 2 𝐽 𝑘 = 0 𝑓 𝐽 , 𝑘 ̂ 𝜃 2 𝐽 𝑥 − 𝑘 , ( 2 . 1 2 ) where ̂ 𝜃 2 𝐽 = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝜃 2 𝑥 − 𝑘 𝐽 + 𝑥 − 𝑘 − 1 𝑛 = − ( 𝐿 − 1 ) 𝑎 𝑛 𝑘 𝜃 2 𝐽 𝜃 2 𝑥 − 𝑛 , 𝑘 = 0 , 1 , … , 2 𝑀 − 1 𝐽 𝑥 − 𝑘 , 𝑘 = 2 𝑀 , … , 2 𝐽 𝜃 2 − 2 𝑀 𝐽 + 𝑥 − 𝑘 2 𝐽 + 𝐿 − 1 𝑛 = 2 𝐽 + 1 𝑏 𝑛 𝑘 𝜃 2 𝐽 𝑥 − 𝑛 , 𝑘 = 2 𝐽 − 2 𝑀 + 1 , … , 2 𝐽 , ( 2 . 1 3 ) where the coefficients 𝑎 𝑛 𝑘 and 𝑏 𝑛 𝑘 are defined by 𝑎 𝑛 𝑘 = 𝑙 1 𝑘 𝑥 𝐽 𝑛 , 𝑏 𝑛 𝑘 = 𝑙 2 𝑘 𝑥 𝐽 𝑛 , ( 2 . 1 4 ) where 𝑙 1 𝑘 ( 𝑥 ) and 𝑙 2 𝑘 ( 𝑥 ) represent Lagrange interpolation polynomials, defined by 𝑙 1 𝑘 ( 𝑥 ) = 2 𝑀 − 1 𝑖 = 0 𝑖 ≠ 𝑘 𝑥 − 𝑥 𝐽 𝑖 𝑥 𝐽 𝑘 − 𝑥 𝐽 𝑖 , 𝑙 2 𝑘 ( 𝑥 ) = 2 𝐽 𝑖 = 2 𝐽 − 2 𝑀 + 1 𝑖 ≠ 𝑘 𝑥 − 𝑥 𝐽 𝑖 𝑥 𝐽 𝑘 − 𝑥 𝐽 𝑖 . ( 2 . 1 5 )

An analogous manner can be given for two-dimensional problem. By using tensor products, it is then possible to define a multiresolution on the square 𝑥 , 𝑦 ∈ [ 0 , 1 ] . The two-dimensional scaling function is defined by Θ 𝐽 𝑘 , 𝑘 ∑ ( 𝑥 , 𝑦 ) = 2 𝐽 𝑘 = 0 ∑ 2 𝐽 𝑘 = 0 𝑓 𝐽 𝑘 , 𝑘 ̂ 𝜃 ( 2 𝐽 ̂ 𝑥 − 𝑘 ) 𝜃 ( 2 𝐽 𝑦 − 𝑘 ′ ) . Let 𝑉 𝐽 𝑥 𝑦 = 𝑉 𝑥 𝐽 ⊗ 𝑉 𝑦 𝐽 be the linear span of the set { 𝜃 ( 2 𝐽 𝑥 − 𝑘 ) 𝜃 ( 2 𝐽 𝑦 − 𝑘 ) , 𝐽 , 𝑘 , 𝑘 ′ ∈ 𝑍 } ; thus the set { 𝑉 𝐽 𝑥 𝑦 } forms a multiresolution analysis and the set { 𝜃 ( 2 𝐽 𝑥 − 𝑘 ) 𝜃 ( 2 𝐽 𝑦 − 𝑘 ) , 𝑘 , 𝑘 ′ ∈ 𝑍 } is a Riesz basis for { 𝑉 𝐽 𝑥 𝑦 } . Therefore, for a function 𝑓 ( 𝑥 , 𝑦 ) defined on 𝑥 , 𝑦 ∈ [ 0 , 1 ] , it has the following approximation: 𝑓 ( 𝑥 , 𝑦 ) = 2 𝐽 𝑘 = 0 2 𝐽 𝑘 = 0 𝑓 𝐽 𝑘 , 𝑘 ̂ 𝜃 2 𝐽 ̂ 𝜃 2 𝑥 − 𝑘 𝐽 𝑦 − 𝑘 ′ . ( 2 . 1 6 )

3. Wavelet Interpolation Galerkin Method for a Parallel Plate Microresonator under the Effect of Squeeze Film Damping

3.1. Governing Equations

In this section, we examine the example of a rectangular parallel plate under the effect of squeeze film damping. As shown in Figure 1, the rectangular parallel plate is excited by a conventional voltage. The voltage is composed of a dc component 𝑉 0 and a small ac component 𝑣 ( 𝑡 ) , 𝑉 0 ≫ 𝑣 ( 𝑡 ) . The plate is rigid. The displacement of the plate under the electric force is composed of a static component to the dc voltage, denoted by 𝑧 0 , and a small dynamic component due to the ac voltage, denoted by 𝑧 ( 𝑡 ) , 𝑧 0 ≫ 𝑧 ( 𝑡 ) , that is, 𝑧 𝐸 ( 𝑡 ) = 𝑧 0 + 𝑧 ( 𝑡 ) . ( 3 . 1 ) The equation of motion that governs the displacement of the plate is written as 𝑚 p l a t e ̈ 𝑧 𝐸 + 𝑘 s p r i n g 𝑧 𝐸 = 𝜀 𝐴 p l a t e 𝑉 0 + 𝑣 2 2 𝑔 0 − 𝑧 𝐸 2 − 𝑓 ( 𝑡 ) , ( 3 . 2 ) where 𝑚 p l a t e is the mass of the plate, 𝐴 p l a t e is the are of the plate, 𝑘 s p r i n g is the stiffness of the spring, 𝑔 0 is the zero-voltage air gap spacing, 𝜀 is the dielectric constant of the gap medium, 𝑓 ( 𝑡 ) is the force acting on the plate owing to the pressure of the squeeze gas film between the plate and the substrate.

fig1
Figure 1: A schematic drawing of an electrically actuated microplate under the effect of squeeze film damping.

We expand (3.2) in a Taylor series around 𝑉 0 and 𝑧 0 up to first order and rewrite (3.2) as 𝑚 p l a t e ̈ 𝑧 + 𝑘 𝐸 𝑧 = 𝜀 𝐴 p l a t e 𝑉 0 ̂ 𝑔 2 0 𝑣 − 𝑓 ( 𝑡 ) , ( 3 . 3 ) where 𝑘 E = 𝑘 s p r i n g − ( 𝜀 𝐴 p l a t e 𝑉 2 0 / ( 𝑔 0 − 𝑧 0 ) 3 ) , ̂ 𝑔 0 = 𝑔 0 − 𝑧 0 . The force 𝑓 ( 𝑡 ) acting on the plate owing to the pressure of the squeeze gas film is given by 𝑓 ( 𝑡 ) = 𝑙 𝑦 0 𝑙 𝑥 0 𝑝 ( 𝑥 , 𝑦 , 𝑡 ) − 𝑝 0 d 𝑥 d 𝑦 , ( 3 . 4 ) where 𝑙 𝑥 and 𝑙 𝑦 are the length and width of plate, 𝑝 ( 𝑥 , 𝑦 , 𝑡 ) is the absolute pressure in the gap and 𝑝 0 is the ambient pressure. The pressure 𝑝 ( 𝑥 , 𝑦 , 𝑡 ) is governed by the nonlinear Reynolds equation [3] 𝜕 ℎ 𝜕 𝑥 3 𝑝 𝜕 𝑝 + 𝜕 𝜕 𝑥 ℎ 𝜕 𝑦 3 𝑝 𝜕 𝑝 𝜕 𝑦 = 1 2 𝜂 e ff ℎ 𝜕 𝑝 𝜕 𝑡 + 𝑝 𝜕 ℎ 𝜕 𝑡 , ( 3 . 5 ) where ℎ ( 𝑥 , 𝑡 ) = 𝑔 0 − 𝑧 0 − 𝑧 ( 𝑡 ) = ̂ 𝑔 0 − 𝑧 ( 𝑡 ) and 𝜂 e ff is the effective viscosity of the fluid in the gap. In this section, all edges of the rectangular plate are ideally vented; thus the pressure boundary conditions for the case in Figure 1 are 𝑝 ( 𝑥 , 0 , 𝑡 ) = 𝑝 𝑥 , 𝑙 𝑦 𝑙 , 𝑡 = 𝑝 ( 0 , 𝑦 , 𝑡 ) = 𝑝 𝑥 , 𝑦 , 𝑡 = 𝑝 0 . ( 3 . 6 )

For convenience, we introduce the nondimensional variables 𝑥 𝑋 = 𝑙 𝑥 𝑦 , 𝑌 = 𝑙 𝑦 𝑧 , 𝑍 = ̂ 𝑔 0 , 𝑝 𝑃 = 𝑝 0 𝑡 , 𝑇 = 𝑆 ℎ , 𝐻 = ̂ 𝑔 0 = 1 − 𝑍 , ( 3 . 7 ) where 𝑇 is a timescale, √ 𝑆 = 𝑚 p l a t e / 𝑘 E = 1 / 𝜔 𝑛 , 𝜔 𝑛 is the nature frequency of the plate. Substituting (3.7) into (3.3)–(3.6), we obtain d 2 𝑍 d 𝑇 2 + 𝑍 = 𝛼 𝑉 0 𝑣 − 𝑃 n o n 1 0 𝑃 − 1 d 𝑋 d 𝑌 , ( 3 . 8 ) 𝜕 𝐻 𝜕 𝑋 3 𝑃 𝜕 𝑃 𝜕 𝑋 + 𝛽 2 𝜕 𝐻 𝜕 𝑌 3 𝑃 𝜕 𝑃 = 𝜎 𝜕 𝑌 𝑆 𝐻 𝜕 𝑃 + 𝑃 𝜕 𝑇 𝜕 𝐻 𝜕 𝑇 , ( 3 . 9 ) where 𝛼 = 𝜀 𝐴 p l a t e / 𝑘 𝐸 ̂ 𝑔 3 0 , 𝑃 n o n = 𝑝 0 𝑙 𝑥 𝑙 𝑦 / 𝑘 𝐸 ̂ 𝑔 0 , 𝜎 = 1 2 𝜂 e ff 𝑙 2 𝑥 / ̂ 𝑔 2 0 𝑝 0 , and 𝛽 = 𝑙 𝑥 / 𝑙 𝑦 . The nondimensional boundary conditions are 𝑃 ( 𝑋 , 0 , 𝑇 ) = 𝑃 ( 𝑋 , 1 , 𝑇 ) = 𝑃 ( 1 , 𝑌 , 𝑇 ) = 𝑃 ( 0 , 𝑌 , 𝑇 ) = 1 . ( 3 . 1 0 )

As mentioned above, the microplate is under small oscillation around ̂ 𝑔 0 and therefore the pressure variation from ambient in the squeeze film is also small, 𝑃 ( 𝑋 , 𝑌 , 𝑇 ) is given by 𝑃 𝑝 ( 𝑋 , 𝑌 , 𝑇 ) = 𝑝 0 = 1 + 𝑃 ( 𝑋 , 𝑌 , 𝑇 ) , ( 3 . 1 1 ) where | 𝑃 ( 𝑋 , 𝑌 , 𝑇 ) | ≪ 1 . Substituting (3.11) into (3.9), and linearizing the outcome around 𝑝 0 and ̂ 𝑔 0 , we obtain 𝜕 2 𝑃 𝜕 𝑋 2 + 𝛽 2 𝜕 2 𝑃 𝜕 𝑌 2 − 𝜎 𝑆 𝜕 𝑃 𝜎 𝜕 𝑇 = − 𝑆 𝜕 𝑍 𝜕 𝑇 . ( 3 . 1 2 ) The boundary conditions for the case are 𝑃 ( 𝑋 , 0 , 𝑇 ) = 𝑃 ( 𝑋 , 1 , 𝑇 ) = 𝑃 ( 0 , 𝑌 , 𝑇 ) = 𝑃 ( 1 , 𝑌 , 𝑇 ) = 0 . ( 3 . 1 3 )

For a harmonic excitation, the ac component voltage v ( t ) is given by 𝑣 ( 𝑇 ) = 𝑣 0 𝑒 𝑗 𝜔 𝑇 𝑆 . ( 3 . 1 4 ) Usually, the excitation frequency 𝜔 is approximate to the natural frequency 𝜔 𝑛 . The steady-state solution of (3.8) and (3.12) may be expressed by 𝑍 ( 𝑇 ) = 𝐴 𝑒 𝑗 𝜔 𝑇 𝑆 , ( 3 . 1 5 ) 𝑃 ( 𝑋 , 𝑌 , 𝑇 ) = 𝐴 ⋅ 𝑃 𝐴 ( 𝑋 , 𝑌 ) 𝑒 𝑗 𝜔 𝑇 𝑆 , ( 3 . 1 6 ) where 𝐴 is the complex amplitude to be determined. Substituting (3.15) and (3.16) into (3.12), we obtain 𝜕 2 𝑃 𝐴 ( 𝑋 , 𝑌 ) 𝜕 𝑋 2 + 𝛽 2 𝜕 2 𝑃 𝐴 ( 𝑋 , 𝑌 ) 𝜕 𝑌 2 − 𝑗 𝜎 𝜔 𝑃 𝐴 ( 𝑋 , 𝑌 ) = − 𝑗 𝜎 𝜔 . ( 3 . 1 7 ) The boundary conditions are 𝑃 𝐴 ( 𝑋 , 0 ) = 𝑃 𝐴 ( 𝑋 , 1 ) = 𝑃 𝐴 ( 0 , 𝑌 ) = 𝑃 𝐴 ( 1 , 𝑌 ) = 0 . ( 3 . 1 8 )

3.2. Wavelet Interpolation Method for Squeeze Film Damping Equations
3.2.1. Construction of Basis Functions

In this subsection, the approximate solution of 𝑃 𝐴 ( 𝑋 , 𝑌 ) is approximated by the following form: 𝑃 𝐴 ( 𝑋 , 𝑌 ) ≈ 2 𝐽 𝑘 = 0 2 𝐽 𝑘 ′ = 0 𝑝 𝐽 𝑘 , 𝑘 ′ Θ 𝐽 𝑘 , 𝑘 ′ ( 𝑋 , 𝑌 ) = 2 𝐽 𝑘 = 0 2 𝐽 𝑘 ′ = 0 𝑝 𝐽 𝑘 , 𝑘 ′ ̂ 𝜃 𝐽 , 𝑘 ̂ 𝜃 ( 𝑋 ) 𝐽 , 𝑘 ′ = ( 𝑌 ) 2 𝐽 𝑘 = 0 2 𝐽 𝑘 ′ = 0 𝑝 𝐽 𝑘 , 𝑘 ′ ̂ 𝜃 2 𝐽 ̂ 𝜃 2 𝑋 − 𝑘 𝐽 𝑌 − 𝑘 , 𝑘 , 𝑘 ∈ 𝑍 , ( 3 . 1 9 ) where the unknowns 𝑝 𝐽 𝑘 , 𝑘 ′ are the values of 𝑃 𝐴 ( 𝑋 , 𝑌 ) at the dyadic points 𝑋 = 𝑘 2 − 𝐽 , and 𝑌 = 𝑘 2 − 𝐽 . The unknowns 𝑝 𝐽 𝑘 , 𝑘 ′ are complex.

For the application of Galerkin method, (3.19) should be able to satisfy the boundary conditions. Substituting (3.19) into (3.18), leads to 2 𝐽 𝑘 ′ = 0 𝑝 𝐽 0 , 𝑘 ′ ̂ ̂ 𝜃 2 𝜃 ( 0 ) 𝐽 𝑌 − 𝑘 = 0 ⟹ 𝑝 𝐽 0 , 𝑘 ′ = 0 , f o r 𝑘 = 0 , 1 , 2 , … 2 𝐽 , 2 𝐽 𝑘 ′ = 0 𝑝 𝐽 2 𝐽 , 𝑘 ̂ ̂ 𝜃 2 𝜃 ( 0 ) 𝐽 𝑌 − 𝑘 = 0 ⟹ 𝑝 𝐽 2 𝐽 , 𝑘 ′ = 0 , f o r 𝑘 = 0 , 1 , 2 , … 2 𝐽 , 2 𝐽 − 1 𝑘 = 1 𝑝 𝐽 𝑘 , 0 ̂ 𝜃 2 𝐽 ̂ 𝑋 − 𝑘 𝜃 ( 0 ) = 0 ⟹ 𝑝 𝐽 𝑘 , 0 2 = 0 , f o r 𝑘 = 1 , 2 , … , 𝐽 , − 1 2 𝐽 − 1 𝑘 = 1 𝑝 𝐽 𝑘 , 2 𝐽 ̂ 𝜃 2 𝐽 ̂ 𝑋 − 𝑘 𝜃 ( 0 ) = 0 ⟹ 𝑝 𝐽 𝑘 , 2 𝐽 2 = 0 , f o r 𝑘 = 1 , 2 , … , 𝐽 . − 1 ( 3 . 2 0 ) Thus (3.19) is rewritten as 𝑃 𝐴 ( 𝑋 , 𝑌 ) = 2 𝐽 − 1 𝑘 = 1 2 𝐽 𝑘 − 1 ′ = 1 𝑝 𝐽 𝑘 , 𝑘 ′ Θ 𝐽 𝑘 , 𝑘 ′ ( 𝑋 , 𝑌 ) = 2 𝐽 − 1 𝑘 = 1 2 𝐽 𝑘 − 1 ′ = 1 𝑝 𝐽 𝑘 , 𝑘 ′ ̂ 𝜃 2 𝐽 ̂ 𝜃 2 𝑋 − 𝑘 𝐽 𝑌 − 𝑘 . ( 3 . 2 1 )

3.2.2. Discretion of the Boundary Value Problem

The weak form functional of (3.17) is 𝑊 𝑃 𝐴 = Ω 1 2 𝜕 𝑃 𝐴 𝜕 𝑋 2 + 𝛽 2 𝜕 𝑃 𝐴 𝜕 𝑌 2 + 𝑗 𝜎 𝜔 𝑃 2 𝐴 − 𝑗 𝜎 𝜔 𝑃 𝐴 d 𝑋 d 𝑌 . ( 3 . 2 2 ) From the necessary conditions for the determination of the minimum 𝑊 , we obtain 𝑃 𝛿 𝑊 𝐴 = Ω 𝜕 𝛿 𝑃 𝐴 𝜕 𝑋 𝜕 𝑃 𝐴 𝜕 𝑋 + 𝛽 2 𝜕 𝛿 𝑃 𝐴 𝜕 𝑌 𝜕 𝑃 𝐴 𝜕 𝑌 + 𝑗 𝜎 𝜔 𝛿 𝑃 𝐴 𝑃 𝐴 − 𝑗 𝜎 𝜔 𝛿 𝑃 𝐴 d 𝑋 d 𝑌 = 0 . ( 3 . 2 3 ) Substituting (3.19) into (3.23), leads to 2 𝐽 − 1 𝑘 = 1 2 𝐽 𝑘 − 1 ′ = 1 Ω 𝜕 Θ 𝐽 𝑚 , 𝑛 𝜕 𝑋 𝜕 Θ 𝐽 𝑘 , 𝑘 ′ 𝜕 𝑋 + 𝛽 2 𝜕 Θ 𝐽 𝑚 , 𝑛 𝜕 𝑌 𝜕 Θ 𝐽 𝑘 , 𝑘 ′ 𝜕 𝑌 + 𝑗 𝜎 𝜔 Θ 𝐽 𝑚 , 𝑛 Θ 𝐽 𝑘 , 𝑘 ′ 𝑝 d 𝑋 d 𝑌 𝐽 𝑘 , 𝑘 ′ = 𝑗 𝜎 𝜔 Ω Θ 𝐽 𝑚 , 𝑛 2 d 𝑋 d 𝑌 , f o r 𝑚 , 𝑛 = 1 , 2 , … , 𝐽 . − 1 ( 3 . 2 4 ) This is a ( 2 𝐽 − 1 ) 2 × ( 2 𝐽 − 1 ) 2 linear system 𝚯 𝐩 = 𝑗 𝜎 𝜔 𝐄 , ( 3 . 2 5 ) where 𝐩 = [ 𝑝 𝐽 1 , 1 𝑝 𝐽 1 , 2 ⋯ 𝑝 𝐽 1 , 2 𝐽 − 1 𝑝 𝐽 2 , 1 𝑝 𝐽 2 , 2 ⋯ 𝑝 𝐽 2 𝐽 − 1 , 2 𝐽 − 1 ] 𝑇 is an ( 2 𝐽 − 1 ) 2 × 1 unknown coefficients’ vector, ∬ 𝐄 = [ Ω Θ 𝐽 1 , 1 ∬ d 𝑋 d 𝑌 Ω Θ 𝐽 1 , 2 ∬ d 𝑋 d 𝑌 ⋯ Ω Θ 𝐽 2 𝐽 − 1 , 2 𝐽 − 1 d 𝑋 d 𝑌 ] 𝑇 is a ( 2 𝐽 − 1 ) 2 × 1 matrix, and Θ is a ( 2 𝐽 − 1 ) 2 × ( 2 𝐽 − 1 ) 2 matrix. The entries in Θ are of the form Θ Θ 2 ( 𝑖 , 𝑗 ) = ( 𝑚 − 1 ) 𝐽 , 2 − 1 + 𝑛 ( 𝑘 − 1 ) 𝐽 − 1 + 𝑘 = Ω 𝜕 ̂ 𝜃 2 𝐽 𝑋 − 𝑚 ̂ 𝜃 2 𝜕 𝑋 𝐽 𝜕 ̂ 𝜃 2 𝑌 − 𝑛 𝐽 𝑋 − 𝑘 ̂ 𝜃 2 𝜕 𝑋 𝐽 𝑌 − 𝑘 + 𝛽 2 ̂ 𝜃 2 𝐽 𝜕 ̂ 𝜃 2 𝑋 − 𝑚 𝐽 𝑌 − 𝑛 ̂ 𝜃 2 𝜕 𝑌 𝐽 𝜕 ̂ 𝜃 2 𝑋 − 𝑘 𝐽 𝑌 − 𝑘 ̂ 𝜃 2 𝜕 𝑌 + 𝑗 𝜎 𝜔 𝐽 ̂ 𝜃 2 𝑋 − 𝑚 𝐽 ̂ 𝜃 2 𝑌 − 𝑛 𝐽 ̂ 𝜃 2 𝑋 − 𝑘 𝐽 𝑌 − 𝑘 d 𝑋 d 𝑌 . ( 3 . 2 6 )

3.2.3. Squeeze Film Damping of the Parallel Plate

The numerical solution of (3.25) can be written as 𝐩 = 𝑗 𝜎 𝜔 𝚯 − 1 𝐄 . ( 3 . 2 7 ) The elements of 𝐩 can be expressed as 𝑝 𝐽 𝑘 , 𝑘 ′ = 𝑝 𝐽 , 𝑅 𝑘 , 𝑘 ′ + 𝑗 𝑝 𝐽 , 𝐼 𝑘 , 𝑘 ′ 2 f o r 𝑚 , 𝑛 = 1 , 2 , … , 𝐽 − 1 , ( 3 . 2 8 ) where 𝑝 𝐽 , 𝑅 𝑘 , 𝑘 ′ and 𝑝 𝐽 , 𝐼 𝑘 , 𝑘 ′ are the real and imaginary parts of 𝑝 𝐽 𝑘 , 𝑘 ′ , respectively. Using (3.21) and (3.28), the force acting on the plate owing to the pressure of the squeeze gas film can be rewritten as 𝑃 n o n 1 0 𝑃 − 1 d 𝑋 d 𝑌 = 𝐴 𝑒 𝑗 𝜔 𝑇 𝑆 ⋅ 𝑃 n o n 2 𝐽 − 1 𝑘 = 1 2 𝐽 𝑘 − 1 ′ = 1 𝑝 𝐽 , 𝑅 𝑘 , 𝑘 ′ + 𝑗 𝑝 𝐽 , 𝐼 𝑘 , 𝑘 ′ 1 0 ̂ 𝜃 2 𝐽 ̂ 𝜃 2 𝑋 − 𝑘 𝐽 𝑌 − 𝑘 d 𝑋 d 𝑌 = 𝐾 𝑎 ⋅ 𝑍 ( 𝑇 ) + 𝐶 𝑎 ⋅ d 𝑍 ( 𝑇 ) , d 𝑇 ( 3 . 2 9 ) where 𝐾 𝑎 = 𝑃 n o n 2 𝐽 − 1 𝑘 = 1 2 𝐽 𝑘 − 1 ′ = 1 𝑝 𝐽 , 𝑅 𝑘 , 𝑘 ′ 1 0 ̂ 𝜃 2 𝐽 ̂ 𝜃 2 𝑋 − 𝑘 𝐽 𝑌 − 𝑘 𝐶 d 𝑋 d 𝑌 , 𝑎 = 𝑃 n o n 𝜔 𝑆 2 𝐽 − 1 𝑘 = 1 2 𝐽 𝑘 − 1 ′ = 1 𝑝 𝐽 , 𝐼 𝑘 , 𝑘 ′ 1 0 ̂ 𝜃 2 𝐽 ̂ 𝜃 2 𝑋 − 𝑘 𝐽 𝑌 − 𝑘 d 𝑋 d 𝑌 ( 3 . 3 0 ) 𝐾 𝑎 ⋅ 𝑍 ( 𝑇 ) and 𝐶 𝑎 ⋅ ( d 𝑍 ( 𝑇 ) / d 𝑇 ) are the spring and damping components of the force. Substituting (3.29), (3.14) and (3.15) into (3.8), we obtain d 2 𝑍 d 𝑇 2 + 𝐶 𝑎 ⋅ d 𝑍 + 𝐾 d 𝑇 𝑎 𝑍 + 1 ( 𝑇 ) = 𝛼 𝑉 0 𝑣 𝑍 ( 𝑇 ) , ( 𝑇 ) = 𝐴 𝑒 𝑗 𝜔 𝑇 𝑆 = 𝑎 𝑉 0 𝑣 0 𝐾 𝑎 ⋅ 1 + 1 𝜔 1 − 2 𝑆 2 / 𝐾 𝑎 + 𝐶 + 1 𝑗 𝜔 𝑎 𝐾 𝑆 / 𝑎 𝑒 + 1 𝑗 𝜔 𝑇 𝑆 , ( 3 . 3 1 ) where √ 𝑆 = 𝑚 p l a t e / 𝑘 𝐸 = 1 / 𝜔 𝑛 . The quality factor and the damped natural frequency are expressed as 𝑄 s q u e e z e = 1 = √ 2 𝜉 𝐾 𝑎 + 1 𝐶 𝑎 , 𝜔 s q u e e z e = 𝜔 𝑛 √ 𝐾 𝑎 + 1 . ( 3 . 3 2 )

4. Wavelet Interpolation Galerkin Method for a Torsion Microplate under the Effect of Squeeze Film Damping

A similar analysis as the one given for the parallel plate microresonator can be given for a torsion microplate.

4.1. Governing Equations

In this section, we examine the example of a rectangular torsion microplate under the effect of squeeze film damping. As shown in Figure 2, the microplate is suspended by two torsion microbeams. 𝑙 𝑥 , 𝑙 𝑦 and ℎ 𝛿 are the length, width and thickness of the plate. There are two pairs of electrodes between the microplate and the substrate. The locations of the two pairs of electrodes are symmetrical. 𝑥 1 and 𝑥 2 are the positions of the two pairs of electrodes. The thickness of the electrodes is neglected. On each pair of the electrodes, an equal dc voltage 𝑉 0 and an equal ac voltage 𝑣 ( 𝑡 ) with opposite potential were applied. The rotation angle of the plate is composed of a static component to the dc voltage, denoted by 𝛾 0 , and a small dynamic component due to the ac voltage, denoted by 𝛾 ( 𝑡 ) . In this case, 𝛾 0 = 0 ; thus the equation of the plate around 𝑉 0 and 𝛾 0 can be written as 𝐽 ̈ 𝛾 + 𝑘 𝑇 − 𝐸 𝑥 𝛾 = − 2 2 − 𝑥 2 1 𝜀 𝑙 𝑦 𝑉 0 𝑔 2 0 𝑣 + 𝑙 𝑥 / 2 − 𝑙 𝑥 / 2 𝑙 0 𝑝 ( 𝑥 , 𝑦 , 𝑡 ) − 𝑝 0 𝑥 d 𝑦 d 𝑥 , ( 4 . 1 ) where 𝑘 𝑇 − 𝐸 = [ 𝑘 𝑇 − 2 𝜀 𝑙 𝑦 𝑉 2 0 ( ( 𝑥 3 2 − 𝑥 3 1 ) / 3 𝑔 3 0 ) ] and 𝑘 𝑇 is the stiffness of the two torsion microbeams. The pressure 𝑝 ( 𝑥 , 𝑦 , 𝑡 ) is governed by (3.5), where ℎ ( 𝑥 , 𝑡 ) = 𝑔 0 + 𝑥 𝛾 ( 𝑡 ) . The pressure boundary conditions for the case in Figure 2 are 𝑝 ( 𝑥 , 0 , 𝑡 ) = 𝑝 𝑥 , 𝑙 𝑦 − 𝑙 , 𝑡 = 𝑝 𝑥 2 𝑙 , 𝑦 , 𝑡 = 𝑝 𝑥 2 , 𝑦 , 𝑡 = 𝑝 0 . ( 4 . 2 )

fig2
Figure 2: A schematic drawing of a torsion microplate under the effect of squeeze film damping.

For convenience, we introduce the nondimensional variables 𝑥 𝑋 = 𝑙 𝑥 + 1 2 , 𝑋 1 = 𝑥 1 𝑙 𝑥 + 1 2 , 𝑋 2 = 𝑥 2 𝑙 𝑥 + 1 2 𝑦 , 𝑌 = 𝑙 𝑦 𝛾 , 𝜗 = 𝛾 m a x , 𝛾 m a x = 2 𝑔 0 𝑙 𝑥 , 𝑝 𝑃 = 𝑝 0 𝑡 , 𝑇 = 𝑆 ℎ , 𝐻 = 𝑔 0 1 = 1 + 2 𝑋 − 2 𝜗 , ( 4 . 3 ) where √ 𝑆 = 𝐽 / ( 𝑘 𝑇 − 𝐸 ) = 1 / 𝜔 𝑛 , 𝜔 𝑛 is the nature frequency of the plate. Substituting (4.3) into (4.1), (3.5) and (4.2), we obtain ̈ 𝜗 + 𝜗 = − 𝛼 𝑉 0 𝑣 + 𝑃 n o n 1 0 1 𝑃 ( 𝑋 , 𝑌 , 𝑇 ) − 1 𝑋 − 2 d 𝑋 d 𝑌 , ( 4 . 4 ) 𝜕 𝐻 𝜕 𝑋 3 𝑃 𝜕 𝑃 𝜕 𝑋 + 𝛽 2 𝜕 𝐻 𝜕 𝑌 3