Analytical modeling of squeeze-film damping for perforated circular microplates

Analytical modeling of squeeze-film damping for perforated circular microplates

Journal of Sound and Vibration 333 (2014) 2688–2700 Contents lists available at ScienceDirect Journal of Sound and Vibration journal homepage: www.e...

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Journal of Sound and Vibration 333 (2014) 2688–2700

Contents lists available at ScienceDirect

Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Analytical modeling of squeeze-film damping for perforated circular microplates Pu Li a,n, Yuming Fang b, Feifei Xu a School of Mechanical Engineering, Southeast University, Jiangning, Nanjing 211189, People0 s Republic of China School of Electronic Science and Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, People0 s Republic of China a

b

a r t i c l e in f o

abstract

Article history: Received 14 September 2013 Received in revised form 25 December 2013 Accepted 28 December 2013 Handling Editor: L. G. Tham Available online 27 January 2014

Predicting squeeze-film damping due to the air gap between the vibrating microstructure and a fixed substrate is crucial in the design of microelectromechanical system (MEMS). The amount of squeeze-film damping can be controlled by providing perforations in microstructures. In the past, to include perforation effects in squeeze-film damping calculations, many analytical models have been proposed. However, only the rectangular perforated microplates are considered in the previous works. There is lack of works that model the squeeze-film damping of circular perforated microplates. In fact, the circular perforated microplates are also common elements in MEMS devices. In this paper, the squeeze-film damping in a perforated circular rigid microplate is modeled using a modified Reynolds equation that includes compressibility and rarefaction effect. The pressure distribution under the vibrating plate is obtained using the Bessel series. Analytical expressions for the squeeze-film damping and spring constants have been found. For a flexible perforated circular microplate, based on Reyleigh’s method, this paper presents an approximate model to estimate the squeeze-film damping in the flexible plate vibrating in the fundamental mode. The accuracy of the present models is verified by comparing its results with the finite element method (FEM) results. & 2014 Elsevier Ltd. All rights reserved.

1. Introduction Squeeze-film damping has been identified as an important mechanism of energy dissipation in micro-resonators. Accurate determination of the squeeze-film damping is very challenging in MEMS area. Over the past two decades, the squeeze-film damping effect on the dynamics of microstructures had been extensively studied. The governing equations of the squeeze-film damping have already been well established. Now, FEM based numerical methods are the most accurate way to evaluate the squeeze-film in MEMS devices. However, many MEMS devices have a simple structure and a simple boundary condition. For the simple structures with the simple boundary conditions, the FEM based numerical methods are cumbersome, time consuming and non-transparent, which is not convenient for the design optimization. In contrast, the analytical models can provide a better insight into the physical characteristic of devices. The aim of this paper is to provide an analytical model for estimating the squeeze-film damping in a perforated circular microplate. Perforations in MEMS structures, often used as etch holes, play a significant role in controlling the squeeze-film damping. To model perforation effects on squeeze-film damping, many methods have been proposed [1–25]. Mohite et al. [1]

n

Corresponding author. Tel.: þ86 25 83552017. E-mail addresses: [email protected], [email protected] (P. Li).

0022-460X/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.12.028

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summarize most of the contributions. Their paper [1] contains an excellent table, summarizing many models and their assumptions. Next, we summarize the analytical models [1–12] for the squeeze-film damping of the perforated microplates. There are two approaches used in the previous analytical models. The basic idea used in both approaches is to consider the vibration plate as a set of uniformly distributed cells. Each cell contains a single hole. In the first approach, the squeezefilm damping within a single cell is calculated by solving the Reynolds equation over the cell using suitable boundary conditions. The total damping is then calculated by multiplying the damping due to the single cell with the total number of cells. In the second approach, the squeeze-film damping and perforation effects are combined in a single modified Reynolds equation. The modified Reynolds equation is solved within the whole plate to directly get the overall damping. The first approach was followed by Skvor [2], Bao et al. [3], Mohite et al. [1,4], Kowk et al. [5], Homentcovschi and Miles [6–8], while the second approach was adopted by Bao et al. [9], Veijola [10], Pandey et al. [11], Pandey and Pratap [12]. As mentioned above, there are many works on analytical modeling of the squeeze-film damping of the perforated microplates. However, only the rectangular perforated microplates are considered in the previous works. Circular perforated microplates are also common elements in MEMS devices. In fact, the squeeze-film damping in the perforated circular microplates can be predicted by the previous models which are based on the first approach. However, the first approach fails to give a reasonable prediction for the microplates with small perforation ratios. The approach is more suitable for large perforation ratios. The previous models which are based on the second approach can give a reasonable prediction over a large range of perforation ratios. However, the previous models based on the second approach are suitable to only deal with the rectangular perforated microplates. From the aforementioned review, we note the lack of works that model the squeeze-film damping in perforated circular microplates. This paper presents an analytical model for calculating the squeeze-film damping in a perforated circular microplate. This paper is based on the second approach. The outline of this paper is as follows. In Section 2, two analytical models for calculating the squeeze-film damping in the rigid plate and the flexible plate are presented respectively. The pressure distribution under the vibrating plate is obtained using the Bessel series. Analytical expressions for the squeezefilm damping and spring constants have been found. Section 3 calculates the squeeze-film damping using the present models, and compares the calculated results with the FEM results. The regime of validity and limitations of the present models are assessed. Finally, a conclusion is given in Section 4. 2. Problem formulation 2.1. Governing equations First we consider a rigid circular microplate under the effect of squeeze-film damping. Fig. 1 shows the example of a rigid circular plate supported by two flexible beams. The two flexible beams can be treated as two springs. The rigid circular plate has a piston-like motion. The rigid circular plate is parallel to the substrate. The rigid circular plate is uniformly perforated with circular holes. Fig. 2 shows the schematic drawing of the perforated circular plate. Tp and R are the thickness and radius of the perforated plate respectively. r0 is the radius of circular holes, N is the total number of circular holes, lp is the pitch of the holes and g0 is the gap spacing. For the rectangular perforated microplate, Bao et al. [9] have derived a modified Reynolds equation under the assumption of incompressible flow for the plate operating at low frequencices by subtracting the pressure relief due to perforations. Pandey et al. [11] have extended Bao’s model to include the compressibility effect and rarefaction effect. In this section, for the circular perforated microplate, the nonlinear modified Reynolds equation presented by Pandey et al. [11] under the isothermal condition can be expressed as   3 Q th β2 r 20 Q ch h 1 ∂ ∂p ∂ðphÞ rp  (1) pðp  pa Þ ¼ ∂r ∂t 12μ r ∂r 8μT eff ηðβÞ where μ is the viscosity of air at ambient conditions, pðr; tÞ is the pressure of air in the gap, pa is the ambient pressure, hðtÞ ¼ g 0 þ A0 ejωt , g 0 is the gap spacing and A0 ejωt is the displacement of the moving plate, T eff ¼ T P þ ð3πr 0 =8Þ is the effective hole length which includes the hole length Tp and an equivalent length to account for the end effect of the hole, β ¼ r 0 =r 1 , r 1 ¼ 0:525lp is the area equivalent outer radius of the pressure cell [1], ηðβÞ ¼ 1 þ ð3r 40 KðβÞQ th =16T eff g 30 Q ch Þ and KðβÞ ¼ 4β2 β4  4 ln β  3, Q ch and Q th are the flow rate factors which account for rarefaction effect in the flow through the parallel

Fig. 1. A schematic drawing of a rigid circular plate supported by two flexible microbeams.

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Fig. 2. A schematic drawing of a perforated circular microplate. (a) Top view of the perforated circular microplate with uniformly distributed holes and (b) cross-sectional view.

plates and through the holes, respectively. The expression for Q ch and Q th are given by pffiffiffi 0:01807 π 1:35355 þ6 1:17468 Q ch ¼ 1 þ 3 D0 D0

(2)

(3) Q th ¼ 1 þ4Knth pffiffiffi where D0 ¼ π=2Knch , Knch ¼ λ=g 0 , Knth ¼ λ=r 0 and λ ¼ 0:0068=pa at ambient temperature and pressure pa. Eq. (1) can be linearized under the assumption of small amplitude vibration ðg 0 bA0 Þ and small pressure variation (pa bΔp), where p ¼ pa þΔp. For convenience, we introduce the following nondimensional variables: Pðr; tÞ ¼

Δp pa

HðtÞ ¼ h0 ejωt

(4)

where h0 ¼ A0 =g 0 . Substituting Eq. (4) into Eq. (1), and linearing the outcome around pa and g0, leads to   ∂2 P 1 ∂P P ∂H 2 ∂P  þ (5) þ ¼ α r ∂r L2 ∂t ∂t ∂r 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where L ¼ 2g 30 T eff ηðβÞQ ch =3β2 r 20 Q th is the characteristic length and α2 ¼ 12μ=Q ch pa g 20 . For the circular plate of radius R which is ideally vented along its edge, the boundary condition is PðR; tÞ ¼ 0

(6)

2.2. Analytical model for the rigid circular plate The displacement of the rigid circular perforated plate is A0 ejωt ; thus the gap variation is HðtÞ ¼ h0 ejωt . In this case, the solution of Eq. (5) can be expressed as M

Pðr; tÞ ¼ ∑ an J 0 n¼1

x  0n r ejωt R

(7)

where an is the complex amplitude to be determined, J 0 ððx0n =RÞrÞ is the Bessel function of zeroth order of first kind, x0n is the nth zero of J 0 ðxÞ. Obviously, J 0 ððx0n =RÞrÞ satisfies the boundary condition (Eq. (6)). Substituting Eq. (7) and HðtÞ ¼ h0 ejωt into

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Eq. (5), leads to   2   M x0n 1 x0n  r ¼ jωα2 h0  2  jωα2 J 0 ∑ an  R R L n¼1

(8)

Multiplying both sides of Eq. (8) by rJ 0 ððx0n =RÞrÞ, and integrating the results from r ¼0 to R leads to an ¼  h0

2α2 jω x0n ½J 1 ðx0n Þ ðx0n =RÞ2 þ ð1=L2 Þ þ jωα2 R

I

¼ a^ n þ ja^ n where

R a^ n

and

I a^ n

(9)

are the real and imaginary parts of an, the expressions of

R a^ n

and

I a^ n

are as follows:

R a^ n ¼ 

A0 2α ω α g 0 x0n ½J 1 ðx0n Þ ½ðx0n =RÞ2 þ ð1=L2 Þ2 þ ω2 α4

(10)

I a^ n ¼ 

A0 2α2 ω½ðx0n =RÞ2 þ ð1=L2 Þ g 0 x0n ½J 1 ðx0n Þ ½ðx0n =RÞ2 þ ð1=L2 Þ2 þ ω2 α4

(11)

2

2 2

As shown in Fig. 1, the equation of motion for the rigid plate can be expressed as ^ ¼ VðtÞ þ F ^ z€ þ kz m Squeeze

(12)

^ and k^ are the mass and stiffness of the device, VðtÞ is a harmonic excitation, F Squeeze is the total force acting on the where m plate owing to the pressure of the squeeze gas film. F Squeeze can be calculated by integrating the pressure distribution on the top surface of the plate Z R Z 2π F Squeeze ¼ ðp pa Þ U rdrdθ (13) 0

0

So the total damping force and the total spring force acting on the plate due to the squeeze-film are Z R Z 2π x  M I 0n F damping ¼ F ISqueeze ¼ pa ∑ a^ n r drdθ rJ 0 R 0 0 n¼1 ¼  A0

4πR2 pa α2 M 1 ω½ðx0n =RÞ2 þð1=L2 Þ ∑ 2 2 2 2 2 4 g0 x n ¼ 1 0n ½ðx0n =RÞ þ ð1=L Þ þ ω α M

R F spring ¼ F RSqueeze ¼ pa ∑ a^ n n¼1

Z 0

R

Z

2π 0

rJ 0

(14)

x  0n r drdθ R

4πR2 pa ω2 α4 M 1 1 ¼  A0 ∑ 2 2 2 2 2 4 g0 n ¼ 1 x0n ½ðx0n =RÞ þð1=L Þ þω α

(15)

where F RSqueeze and F ISqueeze are the real and imaginary parts of F Squeeze . The corresponding damping constant C and the spring constant K owing to the pressure of the squeeze gas film are given by C¼



 F damping 4πR2 pa α2 M 1 ½ðx0n =RÞ2 þ ð1=L2 Þ ¼ ∑ 2 2 2 2 2 4 A0 ω g0 x n ¼ 1 0n ½ðx0n =RÞ þ ð1=L Þ þ ω α

(16)

 F spring 4πR2 pa α2 M 1 ω2 α2 ¼ ∑ 2 2 2 2 2 4 A0 g0 n ¼ 1 x0n ½ðx0n =RÞ þð1=L Þ þω α

(17)

2.3. Analytical approximate model for the flexible circular plate In this subsection, we consider a flexible circular microplate under the effect of squeeze-film damping. Fig. 3 shows the schematic drawing of the flexible circular plate. The geometries of the flexible circular microplate is similar to the rigid circular plate shown in Fig. 2. In MEMS area, the flexible plate is usually operated at the first natural frequency ω1. In this case, we can use Reyleigh’s method to calculate the squeeze-film damping in the flexible circular plate vibrating in the jω1 t ^ fundamental mode. The displacement of a flexible circular microplate is A0 RðrÞe ; thus the gap variation is jω1 t ^ HðtÞ ¼ ðA0 RðrÞ=g Þe . In this work, only the fully clamped circular plate is considered, as the simply supported plates are 0 not often used in MEMS area. For the fully clamped circular plate, we choose the following polynomial function to approximate the deflection function:   2 2 ^ ¼ 1 r RðrÞ (18) 2 R

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Fig. 3. A schematic drawing of a fully clamped flexible circular plate.

Obviously, the chosen function is the static defection function of the fully clamped circular plate under uniform load. The pressure distribution under the vibrating flexible plate can be expressed as x  M 0n Pðr; tÞ ¼ ∑ bn J 0 r ejω1 t (19) R n¼1 where bn is the complex amplitude to be determined. Substituting Eq. (19) and HðtÞ ¼ ðA0 =g 0 Þð1  ðr 2 =R2 ÞÞ2 ejω1 t into Eq. (5), leads to   2    2 M x0n 1 x0n  A0 r2 r ¼ jω1 α2  2  jω1 α2 J 0 1 2 ∑ bn  R R g0 R L n¼1 Multiplying both sides of Eq. (20) by rJ 0 ððx0n =RÞrÞ, and integrating the results from r ¼0 to R leads to " # R I 8 8 ^ ^ 1 bn ¼ bn þ jbn ¼ an 2 x0n x20n " # A0 2α2 jω1 8 8

¼  1 g 0 x0n J 1 ðx0n Þ ðx0n =RÞ2 þ ð1=L2 Þ þ jωU α2 x20n x20n where

(20)

(21)

" # " # ω21 α2 8 A0 2α2 8 8 ^bR ¼ a^ R 8 1 ¼  1 n 2 n g 0 x0n ½J 1 ðx0n Þ ½ðx0n =RÞ2 þ ð1=L2 Þ2 þ ω2 α4 x20n x20n x0n x20n 1

(22)

" # " # I 8 A0 2α2 ω1 ½ðx0n =RÞ2 þ ð1=L2 Þ 8 8 I 8 b^ n ¼ a^ n 2 1 ¼   1 g 0 x0n ½J 1 ðx0n Þ ½ðx0n =RÞ2 þð1=L2 Þ2 þω2 α4 x20n x20n x0n x20n 1

(23)

The equation of the flexible plate vibrating in the fundamental mode can be expressed as m1 q€ þ k1 q ¼ V 1 ðtÞ þ f Squeeze

(24)

^ where m1 and k1 are the first modal mass and stiffness of the structure corresponding to RðrÞ, V 1 ðtÞ is a harmonic excitation R R R 2π ^ p pa rdrdθ is the total modal force acting on the plate owing to the pressure of the (modal force), f Squeeze ¼ 0 0 RðrÞ squeeze gas film. So the total modal damping force and the total modal spring force acting on the plate due to the squeezefilm are 2  Z R Z 2π  M I r2 x0n  I r drdθ f damping ¼ f Squeeze ¼ pa ∑ b^ n r 1  2 J0 R R 0 0 n¼1 " !#2 4πpa α2 R2 M 1 ω1 ½ðx0n =RÞ2 þð1=L2 Þ 8 8 ¼  A0 ∑ 2 1 (25) 2 2 2 2 2 4 x2 g0 n ¼ 1 x0n ½ðx0n =RÞ þ ð1=L Þ þ ω1 α 0n x0n  2  r2 x0n  r drdθ r 1  2 J0 R R 0 0 n¼1 " !#2 ω21 α2 4πpa α2 R2 M 1 8 8 ¼  A0  1 ∑ 2 2 2 2 2 2 4 x2 g0 n ¼ 1 x0n ½ðx0n =RÞ þ ð1=L Þ þ ω1 α 0n x0n M

R R f spring ¼ f Squeeze ¼ pa ∑ b^ n

Z

R

Z



(26)

The corresponding modal damping constant Cf and spring constant Kf owing to the pressure of the squeeze gas film are given by " !#2 f damping 4πR2 pa α2 M 1 ½ðx0n =RÞ2 þ ð1=L2 Þ 8 8 Cf ¼ ¼ ∑ 2 1 (27) 2 2 2 2 2 4 x2 A0 ω1 g0 n ¼ 1 x0n ½ðx0n =RÞ þ ð1=L Þ þ ω1 α 0n x0n

P. Li et al. / Journal of Sound and Vibration 333 (2014) 2688–2700

" f spring ω21 α2 4πR2 pa α2 M 1 8 Kf ¼ ¼ ∑ 2 h i2 2 2 A0 g0 x 2 2 n ¼ 1 x0n x0n =R þ ð1=L Þ þ ω1 α4 0n

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8 1 x20n

!#2 (28)

The corresponding modal damping ratio ξ1 is calculated from the following expressions: ξ1 ¼

Cf 2ω1 m1

(29)

RR 2 where m1 is the modal mass corresponding to the first natural frequency ω1 , given by m1 ¼ 2πρperforated T P 0 r R^ ðrÞdr ¼ 0:2πρperforated T P R2 , ρperforated ¼ ρplate ðR2 Nr 20 =R2 Þ and ρplate is the density of the plate material. In this section, only the analytical approximate model for the squeeze-film damping of the circular plate vibrating at the first mode has been presented. The analytical approximate models for the circular plate vibrating at higher modes have not been provided. The main reasons are as follows. The most important issue in the Rayleigh method is the selection of the shape functions. For the perforated circular plate vibrating at the first mode, we choose an appropriate polynomial function (Eq. (18)) to approximate the deflection function. In fact, the exact first mode shape of the non-perforated microplate can also be used as the shape function. However, in MEMS area, the flexible microplate is excited by the electrostatic force. The electrostatic force in the case of small deflection is very close to the uniformly distributed. So we choose Eq. (18) to approximate the deflection function. The function (Eq. (18)) is the static deflection function of the fully clamped circular plate under uniform load. Eq. (18) is more closer to the actual deflection of the micro plate than the exact mode shape. For the perforated circular plate vibrating at higher modes, it is very difficult to select an appropriate polynomial function to approximate the deflection function. The exact mode shapes of the circular plate vibrating at higher modes cannot also be used as the shape function. The analytical expression of the exact mode shapes is only valid for the circular plate with no perforations. For the perforated circular plate vibrating at the first mode, the ANSYS results show that the discrepancy between the first mode shape of the non-perforated circular plate and the actual deflection of the perforated circular plate increases as perforation ration increases. In fact, the discrepancy between the mode shape of the non-perforated circular plate and the actual deflection of the perforated circular plate also increases as mode increases. For the perforated circular plate vibrating at higher modes with large perforation ratio, the discrepancy between the actual deflection (ANSYS result) and the mode shape of the non-perforated circular plate is very significant. Therefore it is unable to use the exact mode shapes as the shape function.

3. Validation and discussions 3.1. Comparsions with the FEM results for the rigid circular microplates In this section, we use the present model to predict the squeeze-film damping in the rigid perforated circular plates at different values of perforation ratios (2r 0 =lp ), and then compare these predictions with the FEM results. The circular plate with 19 holes shown in Fig. 2 is considered in this section. The dimensions and parameters of the microplates used in simulations are listed in Table 1. The amplitude of the vibrating plate is very small (A0 ¼ g 0 =100). The FEM simulations are performed using ANSYS. In ANSYS, the squeeze-film effect due to the fluid film of the plate is modeled using four-noded FLUID136 elements. The finite resistance due to the fluid flow through the holes is modeled using two-noded FLUID138 elements. One end of the FLUID138 element is connected to all the nodes of FLUID136 lying on the circumference of the hole and the pressure is set to zero on the free node at the other end of hole. Zero pressure boundary conditions are applied on the free boundaries of the plate. Fig. 4 illustrates a typical ANSYS model with boundary conditions. For comparison purposes, we also give the results obtained by Homentcovshi’s model [8]. Homentcovshi’s model is based on the first approach, i.e., the model is based on the solution of Reynolds equation under a single hole-cell combination with non-trivial pressure boundary condition on the inner boundary and trivial pressure boundary condition on the outer boundary of the cell. The total damping is obtained by multiplying the damping contribution of this cell with the total number of cells. By adding the Table 1 The dimensions and parameters of the microplates used in the simulations. Symbol

Description

Values

Unit

R Tp lp N g0 pa μ

Radius of the circular microplate Thickness of the circular microplate The pitch of the holes The total number of the holes Gap spacing Ambient pressure Viscosity coefficient

125 5 50 19 5 1.013  105 1.83  10  5

mm mm mm mm N/m2 N s/m2

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One end of the FLUID138 element is connected to all the nodes of FLUID136 lying on the circumference of the hole and the pressure is set to zero on the free node at the other end of hole.

Zero pressure boundary conditions are applied on the free boundaries of the plate.

Fig. 4. ANSYS model with boundary condition for the perforated rigid micrplate (2r0/lp ¼ 0.5).

The imaginary part of the pressures (N/m2)

1 The pressure with M = 10 The pressure with M = 5 The pressure with M = 1

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

r/R

6

x 10-4

The real part of the pressures (N/m2)

The pressure with M = 10 The pressure with M = 5 The pressure with M = 1

5

4

3

2

1

0

0

0.2

0.4

r/R

0.6

0.8

1

Fig. 5. Convergence analysis of the series for the pressures in the gap. These results are for the microplate with 2r0/lp ¼0.1. (a) Comparison of the imaginary parts of the pressures in the cases of M ¼1, 5 and 10 and (b) comparison of the real parts of the pressures in the cases of M¼ 1, 5 and 10.

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The imaginary part of the pressures (N/m2)

8

2695

x 10-3 The pressure with M = 20 The pressure with M = 10 The pressure with M = 5 The pressure with M = 1

7 6 5 4 3 2 1 0

0

0.2

0.4

0.6

0.8

1

r/R

The real part of the pressures (N/m2)

3.5

x 10-8 The pressure with M = 20 The pressure with M = 10 The pressure with M = 5 The pressure with M = 1

3 2.5 2 1.5 1 0.5 0

0

0.2

0.4

0.6

0.8

1

r/R Fig. 6. Convergence analysis of the series for the pressures in the gap. These results are for the microplate with 2r0/lp ¼0.8. (a) Comparison of the imaginary parts of the pressures in the cases of M¼ 1, 5, 10 and 20 and (b) comparison of the real parts of the pressures in the cases of M¼ 1, 5, 10 and 20.

resistance of holes, the expression of damping constants proposed by Homentcovschi and Miles [8] is C Homent ¼ N

3π μ r 41 μ r 41 KðβÞ þN8πT P 3 2 Q ch g 0 Q th r 40

(30)

where the first term captures the loss due to squeeze-film flow under the cells and the second term captures the additional loss due to the resistance of holes, r 1 ¼ 0:525lp is the area equivalent outer radius of the pressure cell. Homentcovshi’s model is valid only for the incompressible fluids. Therefore a frequency of 500 Hz is used in our simulation. In this case, the spring force is much less than the damping force. The infinite summations in Section 2.2 for squeeze-film damping were evaluated using MATLAB. Careful convergence studies were performed. Figs. 5 and 6 show the rate of convergence for two microplates with 2r 0 =lp ¼0.1 and 0.8 respectively. Obviously, for the microplate with small perforation ratio (2r 0 =lp ¼0.1), a total of 5 terms were sufficient to achieve convergence. As the results for n¼ 5 are indistinguishable from the results for n¼10. The discrepancy between the results for n¼ 1 and 5 is insignificant. However, for the microplate with large perforation ratio (2r 0 =lp ¼0.8), a total of 20 terms were sufficient to achieve convergence. The discrepancy between the results for n¼ 1, 5 and 10 is significant. For this, and all other cases presented in this paper, a total of 20 terms were used to predict the squeeze-film damping. Fig. 7 shows the damping constants of the microplates as a function of perforation ratio (2r 0 =lp ) obtained by the FEM model and the present model (Eq. (16)). Also shown in the same figure are the results obtained by Homentcovshi’s model

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10-3 FEM results Homentcovshi’s model The present model

The damping constant (N·s/m)

10-4

10-5

10-6

10-7

10-8

0

0.2

0.4

0.6

0.8

1

Perforation ration (2r0/lp) Fig. 7. Comparison of the damping constant obtained by the FEM model, the present model and Homentcovshi0 s model [8].

10-3 FEM results The present model

The spring constant (N/m)

10-4

10-5

10-6

10-7

10-8

10-9

0

0.2

0.4

0.6

0.8

1

Perforation ration (2r0/lp) Fig. 8. Comparison of the spring constant obtained by the FEM model and the present model.

(Eq. (30)). For the analytical models, we varied perforation ratios from 0.01 to 0.99 for the same values of pitch (lp ¼50 mm) and other dimensions as mentioned above. For the FEM model, we varied perforation ratios from 0 to 0.9. As expected, the present model gives results in good agreement with the FEM results. The present model performs much better than Homentcovshi’s model. Obviously, in Homentcovshi’s model, C Homent -1 in the case of r0 ¼0. Homentcovshi’s model fails to give a reasonable prediction for the microplates with small perforation ratios. Fig. 8 shows the comparison of spring constants of the microplates obtained by the FEM model and the present model (Eq. (17)). In the range from 2r 0 =lp ¼0 to 0.6, the present model gives results in good agreement with the FEM results. However, above 2r 0 =lp ¼0.7, the discrepancy between the FEM model and the present model increases as the perforation ratio increases. For the plates with 2r 0 =lp ¼0.7, 0.8 and 0.9, the discrepancies between the FEM model and the present model are 32.5 percent, 53.7 percent and 69.2 percent respectively. In order to check the validity of the present model at different values of frequency, we compare the damping forces and the spring forces obtained by the present model and the FEM model in the range from 1 kHz to 1000 MHz. It needs to be emphasized that the inertia terms are not considered in the present model. The inertia terms can be neglected where the modified Reynolds number Re ¼ ρair ωg 20 =μ⪡1. This sets an upper limit for the microplates on the maximum frequency. In our simulation, the maximum frequency for Re ¼0.1 is about 10 kHz. Therefore it is unnecessary to compare the results above

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10 kHz. Above 10 kHz, the inertia terms cannot be neglected. However, to examine the validity of the present model over a wide range of frequency, we also compare the results obtained by the present model with the FEM results above 10 kHz. In fact, the inertia terms are also not included in the FLUID136 element of ANSYS. Figs. 9–11 show the damping forces and the spring forces obtained by the two models for the microplates at 2r 0 =lp ¼0.2, 0.5 and 0.8 respectively. It can be seen that the match between the present model and the FEM results is very good over a wide range of frequency for the microplates with 2r 0 =lp ¼0.2 and 0.5. The discrepancy between the present model and the FEM model increases as the perforation ratio increases. For the microplates with large perforation ratios (2r 0 =lp ¼0.8), the maximum deviation in the damping force and the spring force below 107 Hz is about 14.4 percent and 53.7 percent respectively. Above 107 Hz, the maximum deviation in the damping force and the spring force is about 224.7 percent and 125.1 percent respectively.

10-4 10-5

Force (N/m)

10-6 10-7 10-8 10-9 10-10

Damping force (The present model)

10-11

Damping force (FEM results) Spring force (The present model) Spring force (FEM results)

10-12 3 10

104

105

106

107

108

109

Frequency (Hz) Fig. 9. Comparison of the damping forces and the spring forces for microplate (2r0/lp ¼0.2) obtained by the FEM model and the present model as a function of frequency.

10-4 10-5 10-6

Force (N/m)

10-7 10-8 10-9 10-10 10-11

Damping force (The present model) Damping force (FEM results) Spring force (The present model) Spring force (FEM results)

10-12 10-13

103

104

105

106

107

108

109

Frequency (Hz) Fig. 10. Comparison of the damping forces and the spring forces for microplate (2r0/lp ¼0.5) obtained by the FEM model and the present model as a function of frequency.

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3.2. Comparsions with the FEM results for the flexible circular microplates Now we use the present model to predict the squeeze-film damping in the flexible circular perforated microplates at different values of perforation ratios (2r 0 =lp ), and then compare these predictions with the FEM results. The thickness of the microplates is 2 mm and other dimensions as mentioned in Table 1. The microplates have the following material properties: E ¼165 GPa, v ¼0.23 and ρplate ¼ 2330 kg/m3. In ANSYS, Modal projection technique is used to calculate the modal damping ratio and the modal stiffness of the microplates vibrating in the fundamental model. The FLUID136 element is used to model the fluid domain between a fixed surface and a structure moving normally to that surface. The FLUID138 element is used to model the pressure drop through holes in a moving structure. The structure of the microplate is modeled with SOLID185 elements.

10-4 10-5 10

Force (N/m)

10-7 10-8 10-9 10-10 10-11 10-12

Damping force (The present model) Damping force (FEM results) Spring force (The present model)

-13

10

Spring force (FEM results)

10-14

103

104

105

106

107

108

109

Frequency (Hz) Fig. 11. Comparison of the damping forces and the spring forces for microplate (2r0/lp ¼0.8) obtained by the FEM model and the present model as a function of frequency.

FEM results

The first modal damping ratio

10-1

The present model

10-2

10-3

10-4

0

0.2

0.4

0.6

0.8

1

Perforation ration (2r0/lp) Fig. 12. Comparison of the first modal damping ratio obtained by the FEM model and the present model.

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1013

FEM results The present model

The first modal spring stiffness

1013

1012

1011

109

108

107

106

0

0.2

0.4

0.6

0.8

1

Perforation ration (2r0/lp) Fig. 13. Comparison of the first modal stiffness obtained by the FEM model and the present model.

Table 2 Comparison of the first modal damping ratio and the first modal stiffness obtained by the FEM results and the present model. 2r 0 lp

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

f1 (Hz)

5.428  105 5.384  105 5.325  105 5.287  105 5.300  105 5.201  105 5.102  105 4.987  105 4.901  105 4.884  105

The first modal damping ratio

The first modal stiffness (K f =m1 )

FEM results

The present model (error)

FEM results

The present model(error)

8.824  10  2 7.694  10  2 3.634  10  2 1.767  10  2 9.734  10  3 5.835  10  3 3.395  10  3 1.827  10  3 7.866  10  4 1.496  10  4

8.829  10  2 7.964  10  2 3.960  10  2 1.991  10  2 1.153  10  2 7.241  10  3 4.584  10  3 2.872  10  3 1.790  10  3 1.220  10  3

1.517  1012 9.400  1011 1.625  1011 3.684  1010 1.179  1010 4.102  109 1.372  109 3.978  108 8.662  107 7.521  106

1.552  1012 (0.0%) 1.006  1012 (7.0%) 1.846  1011 (13.6%) 4.193  1010 (13.8%) 1.327  1010 (12.6%) 4.462  109 (8.8%) 1.483  109 (8.1%) 4.591  108 (15.4%) 1.361  108 (57.1%) 4.667  107 (520.5%)

(0.0%) (3.5%) (8.9%) (12.7%) (18.5%) (24.1%) (35.0%) (57.2%) (127.6%) (715.4%)

Figs. 12 and 13 show the comparisons of the modal damping ratio and the modal stiffness obtained by the FEM model and the present model. Obviously, the present model gives results in good agreement with the FEM results for the plates with the smaller and medium perforation ratios. For the first modal damping ratio, in the case of smaller and medium perforation ratios (2r 0 =lp r 0:6), the present model gives values very close to the FEM results. For the first modal stiffness, below 2r 0 =lp ¼ 0:7, the present model gives values very close to the FEM results. The discrepancy between the FEM model and the present model increases as the perforation ratio increases. Table 2 lists the detailed values and the percentage differences between the FEM results and the present model, using the FEM results as baseline. 4. Conclusions This paper presents an analytical model for calculating the squeeze-film damping in a rigid perforated circular microplate. The pressure distribution under the vibrating plate is obtained using the Bessel series. Analytical expressions for the squeeze-film damping and spring constants have been found. For the flexible perforated circular microplate, based on Reyleigh’s method, this paper presents an approximate model to estimate the squeeze-film damping in the flexible plate vibrating in the fundamental mode. On comparing the present model with the FEM results, the following conclusions can be drawn. The present model is more suitable for the plates with smaller and medium perforation ratios. For the rigid perforated circular microplates, in the case of smaller and medium perforation ratios (2r 0 =lp r0:6), the present model gives results in good agreement with the FEM results over a wide range of frequency. For the microplates with large perforation ratios

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(2r 0 =lp Z 0:7), deviations between the present model and the FEM model for the damping constants are observed only at higher values of operating frequencies. However, for the spring constants, there is a clear discrepancy between the two models over a wide range of frequency. For the flexible perforated circular microplates operated at the fundamental mode, the present model also gives results in good agreement with the FEM results in the case of smaller and medium perforation ratios. For the devices with higher values of air gap and operating frequencies, the Reynolds number would be higher, and a model considering inertial effects has to be developed. Acknowledgments This project is supported by National Natural Science Foundation of China (Grant no. 51375091), Jiangsu Provincial Natural Science Foundation of China (Grant no. BK20131380) and Qing Lan Project in Jiangsu Provincial. References [1] S.S. Mohite, V.R. Sonti, R. 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