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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO. 3, JUNE 2008 709

A Compact Squeeze-Film Model Including Inertia,Compressibility, and Rarefaction Effects for

Perforated 3-D MEMS StructuresSuhas S. Mohite, Venkata R. Sonti, and Rudra Pratap, Member, IEEE

Abstract—We present a comprehensive analytical model ofsqueeze-film damping in perforated 3-D microelectromechanicalsystem structures. The model includes effects of compressibility,inertia, and rarefaction in the flow between two parallel platesforming the squeeze region, as well as the flow through perfora-tions. The two flows are coupled through a nontrivial frequency-dependent pressure boundary condition at the flow entry in thehole. This intermediate pressure is obtained by solving the fluidflow equations in the two regions using the frequency-dependentfluid velocity as the input velocity for the hole. The governingequations are derived by considering an approximate circularpressure cell around a hole, which is representative of the spatiallyinvariant pressure pattern over the interior of the flow domain.A modified Reynolds equation that includes the unsteady inertialterm is derived from the Navier–Stokes equation to model theflow in the circular cell. Rarefaction effects in the flow throughthe air gap and the hole are accounted for by considering theslip boundary conditions. The analytical solution for the net forceon a single cell is obtained by solving the Reynolds equationover the annular region of the air gap and supplementing theresulting force with a term corresponding to the loss through thehole. The solution thus obtained is valid over a range of air gapand perforation geometries, as well as a wide range of operatingfrequencies. We compare the analytical results with extensivesimulations carried out using the full 3-D Navier–Stokes equationsolver in a commercial simulation package (ANSYS-CFX). Weshow that the analytical solution performs very well in trackingthe net force and the damping force up to a frequency f = 0.8fn

(where fn is the first resonance frequency) with a maximumerror within 20% for thick perforated cells and within 30% forthin perforated cells. The error increases considerably beyond thisfrequency. The prediction of the first resonance frequency is within21% error for various perforation geometries. [2007-0001]

Index Terms—Gyroscopes, microactuators, microphones,microresonators, microsensors.

I. INTRODUCTION

E FFICIENT compact models for the squeeze-film analysisof perforated microelectromechanical system (MEMS)

structures are indispensable in the design of MEMS devices.The most important design goals for dynamic MEMS devicessuch as accelerometers, microphones, tuning-fork gyroscopes,microswitches, micromirrors, etc., are high sensitivity and high

Manuscript received January 1, 2007; revised February 29, 2008. This workwas supported by a grant from the Department of Science and Technology,Government of India. Subject Editor G. Fedder.

The authors are with the CranesSci MEMS Laboratory, Department ofMechanical Engineering, Indian Institute of Science, Bangalore 560012, India.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JMEMS.2008.921675

resolution. When designing MEMS parallel-plate capacitivetransducers for these applications, the aforementioned goalscan be achieved by maximizing the mechanical complianceCm ∝ (A/t3) and the base capacitance Cb ∝ (A/ha) of thestructures. This clearly requires sufficiently high surface area Aand small thickness t of the vibrating structure, as well as avery small air-gap ha separating the two structures. However,the air in the small gap between the transversely moving planarstructure and the fixed substrate imparts damping, spring, andinertial forces to the structures [1]–[3]. These forces have acomplex dependence on the air-gap height and the operatingfrequency. These forces affect the frequency response of thestructure and, hence, the sensitivity, resolution, and bandwidthof the device. Damping due to the squeeze-film is considerablyhigh for large surface-to-air-gap ratio. At low frequencies,damping dominates, whereas at high frequencies, spring andinertial effects dominate. Damping can be minimized if it ispossible to operate the device under a vacuum condition, which,in turn, requires expensive packaging. However, in cases withhigh-Q materials, such as silicon, the Q-factor of a vibratingsystem is still mainly determined by the energy losses tothe surrounding air as the vacuum in the encapsulated devicecan hardly be high enough [4]. The amount of squeeze-filmdamping can be controlled by providing perforations in eitherthe back plate or the oscillating proof mass. These perforationsalso facilitate the etch release of the sacrificial layer in surfacemicromachining but reduce the device capacitance, which isundesirable. An adequate value of the base capacitance can beobtained by reducing the air-gap. While reducing the gap, theperforations need to be designed such that constant dampingis maintained without sacrificing the capacitance. A feasiblerange of perforation geometry depends on the chosen micro-machining process. To achieve the design goals of minimizingdamping and maximizing the capacitance, a tradeoff analysismust be carried out between the perforation geometry and theair-gap. This necessitates creation of simple analytical modelsand derivation of closed-form formulas for the squeeze-filmanalysis. These models, in turn, should be validated using finite-element method (FEM) tools or experimental measurements.The forces due to the squeeze-film extracted from these behav-ioral models are, in turn, employed in a system-level model forthe performance optimization of a MEMS device. Thus, the air-gap ha is perhaps the single most crucial parameter that affectsthe squeeze-film behavior, the base capacitance, and, hence, theoverall device performance. However, for small air-gap, therarefaction and compressibility effects are predominant. In

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710 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO. 3, JUNE 2008

addition, at higher frequencies, the compressibility and inertialeffects considerably increase and cannot be neglected. In allsuch applications, where these effects cannot be neglected,the proposed model will be very valuable. Such applicationsinclude high-frequency MEMS transducers such as ultrasoundtransducers (with perforated back plates), moderately high fre-quency devices with larger air-gap, etc. Although obvious, weadd here that the proposed model can, of course, be used inregular cases as well, where compressibility and inertia arenegligible.

In the next section, we discuss various phenomena associatedwith the squeeze-film effect in terms of the nondimensionalnumbers characterizing them and highlight how they are relatedto the perforation geometry and the operating frequency of adevice.

II. MODELING RAREFACTION, COMPRESSIBILITY,AND INERTIAL EFFECTS

A. Rarefaction

The degree of rarefaction in the fluid flow is characterized bythe Knudsen number Kn. For the air-gap, we define Kng =λ/ha, where ha is the air-gap thickness, and λ is the meanfree path of the gas molecules. For the flow through a hole,we define Knh = λ/ri, where ri is the radius of the hole. Themolecular mean free path λ is related to the packaging pressureP as λ = Paλa/P . At ambient conditions, the mean free pathλa for air is 0.064 µm. For MEMS devices with very small air-gap and low packaging pressure, the Knudsen number becomessufficiently high, and the fluid flow transits from a continuumto a rarefied flow. Based on the values of Kn, the flow regimesare usually divided into four different types: continuum (Kn <0.01), rarefaction (0.01 < Kn < 0.1), transition (0.1 < Kn <3), and molecular (Kn > 3) regimes [5]. The relative flow ratecoefficient Qpr(h, p) is generally included in the fluid viscosity,and the combined term is called the effective viscosity µeff =µ/Qpr. Veijola et al. [6] present a simple approximation for theeffective viscosity by fitting the respective flow rate coefficientsto the experimental values tabulated by Fukui and Kaneko [7].This relationship is given as µeff = µ/(1 + 9.638Kn1.159

g ).The value of µeff drops by 30% (from µ) for a 1-µm air-gapat 1-atm pressure. For the slip-flow regime, an analytical modelis derived using the slip-flow wall boundary conditions on theplanar surfaces in the air-gap and on the cylindrical on thehole walls.

B. Compressibility

The repetitive patterns of pressure distribution observedaround the etch holes in a perforated structure can be approx-imated by equivalent circular pressure cells with a ventingboundary condition on the edge of the holes. Consideration ofsuch circular cells considerably simplifies the flow analysis.In the squeeze-film analysis, the extent of compressibility ismeasured by the squeeze number σ. An expression for thesqueeze number for an annular plate is given in [8] as σ =12µωr2

o/Pah2a((1 − β)2/π2), where β is the ratio of the inner

radius to the outer radius of the annulus with a value near

unity. A more accurate expression for the squeeze number isderived in [9] by considering the squeeze-film flow, as well asthe flow through holes and various other losses. The commonparameters in these expressions and in the analytical expres-sions derived in this paper are given by 12µωr2

o/Pah2a. Here,

ro is the radius of an equivalent circular pressure cell, whichis approximately half the pitch of the holes (ro ≈ ξo/2). Itis observed that compressibility is proportional to the squareof the pitch-to-air-gap ratio (ξo/ha)2 and the frequency ofoscillation ω, and it is inversely proportional to the ambientpressure Pa. If σ � 1, the compressibility can be neglected,and the flow can be treated as incompressible. For higher valuesof σ, the compressibility leads to a significant air-spring effect,which can be undesirable as it can adversely affect the dynamicbehavior of a device [10]. Contrary to this, the fact that perfo-rations increase the cutoff frequency—the frequency at whichthe damping and spring forces are equal—can be exploited bysuitably tuning both of these forces by varying the numberand size of perforations [11]. Blech [12] and Allen et al. [13]have also reported the use of squeeze-film damping to tailorthe frequency response of a seismic accelerometer and that ofmicromachined sensors, respectively. Therefore, including thecompressibility effects in the analysis enables the designer todesign the perforation geometry such that the compressibilityeffects are either totally ruled out or suitably tuned, as thecase may be.

C. Inertia

The small dimensions of MEMS devices lead to a very smallvolume, which contains a minuscule quantity of air. Hence,fluid inertia may be neglected at low frequencies (Re � 1).However, for larger air-gap heights and at higher frequenciesof oscillations, the inertial effects may not be negligible and,hence, need to be considered when calculating the qualityfactor of such devices [3]. This is done by incorporating thefrequency-dependent flow rate coefficient Qpr, which modifiesthe velocity profile with frequency. For the squeeze-film flowin the air-gap, the modified Reynolds number is defined asReg = ρωh2

a/µ, and for the flow through the hole, it is definedas Reh = (ρωr2

i )/µ. As is evident, the Reynolds number in thetwo cases increases with the square of the air-gap height andthe radius of the hole, respectively.

The analytical model presented in this paper incorporatescompressibility, inertia, and gas rarefaction effects and is ap-plicable over a moderate range of perforation geometries and awide range of frequencies. In the next section, we briefly reviewvarious modeling and analysis approaches used for squeeze-film damping of perforated structures.

III. MODELING STRATEGIES FOR

PERFORATED STRUCTURES

The squeeze-film flow in dynamic MEMS structures is wellmodeled by the 2-D Reynolds equation obtained from theNavier–Stokes equation by neglecting inertia and assumingthe lateral dimension to be an order of magnitude largerthan the air-gap height. For perforated MEMS structures, the

MOHITE et al.: SQUEEZE-FILM MODEL INCLUDING INERTIA, COMPRESSIBILITY, AND RAREFACTION EFFECTS 711

effective lateral dimension is the pitch of the holes ξo. There-fore, in the case of perforated structures, the assumption of alarge lateral dimension begins to deviate. Moreover, the flowthrough the perforations comes into play, adding an extra di-mension to the model. The model dimensionality (2-D or 3-D)is governed by the relative size of the perforation parameters.The significant perforation parameters are the pitch ξo, thediameter d, and the length of the holes l (which is the same asthe thickness of the perforated plate). Various models have beenpresented for the squeeze-film analysis of perforated MEMSstructures in the literature. These models differ in variousrespects such as the model dimensionality, the methodologiesused, and the different effects considered in the governingequations. Fig. 1 gives an overview of the different perforationgeometries and various squeeze-film models at a glance. So-lution methods based on numerical, analytical, or mixed tech-niques are developed to suit the different perforation geometriesshown in Fig. 1(a). Numerical methods are best suited formodeling the nonlinear pressure response due to the high am-plitudes of oscillation, arbitrary boundary conditions, complexgeometries, and nonuniform size and distribution of the holes[10], [15], [16]. Computationally efficient FEM-based simula-tion schemes are reported in the recent past [17]–[22]. A hybridmodel combining the Navier–Stokes equation and the Reynoldsequation is proposed in [17]. A mixed-level approach based onFEM and finite network simulation is presented in [18]–[20].A hierarchical two-level simulation strategy is employed, anda coupled reduced dimensional analysis is performed in [21].Another method for the arbitrary perforation problem utilizesa Perforation Profile Reynolds solver, which is a multiphysicssimulation software [22]. These seminumerical approaches aremore efficient than the numerical methods but take time to buildthe model and require special computational tools.

Our goal of modeling and analyzing the squeeze-film behav-ior in perforated structures is to perform a reasonably accurateanalysis with minimal modeling and computational effort. Inthis respect, analytical methods are desirable since they giveclosed-form expressions that can directly be used in system-level simulations to evaluate design tradeoffs. In these methods,the repetitive pressure pattern around each hole is exploited, andthe entire perforated domain is discretized into pressure cellsunder simplifying assumptions [9], [23]–[31].

We now discuss different analytical models used for variousperforation geometries, as shown in Fig. 1(a.1)–(a.3). When thesize of the holes is much larger compared to the air-gap height(d > 10ha) and the perforations are short (l/d < 1), as shownin Fig. 1(a.1), the loss through the holes is not significant. In thiscase, the problem is reduced to solving the 2-D Reynolds equa-tion under the simplified boundary condition that the acousticpressure vanishes at the edge of a hole. Closed-form solutionsderived assuming incompressible flow exist [23], and solutionsincluding rarefaction, compressibility, and inertial effects arealso reported [27]–[29].

For inertial MEMS sensors (e.g., gyroscope and accelerome-ter), high-aspect-ratio (l/d > 1) perforated structures using adeep reactive-ion etching process are designed, as shown inFig. 1(a.2). In this case, the loss through holes can be significantas the flow is a combination of the horizontal flow between the

planar surfaces and the vertical flow through the holes. Thisproblem is modeled by modifying the Reynolds equation byadding a pressure leakage term corresponding to the loss due tothe flow through the holes (in the z-direction) [24]–[26]. It isassumed that for uniformly distributed holes, the contributionof holes can be homogenized over the entire domain. In thesemodels, the flow is assumed to be incompressible. Alternatively,the flow problem can be treated in two distinct parts. Thelateral flow is modeled by the Reynolds equation, and the flowthrough the holes is modeled by the Poiseuille equation, asdone by Homentcovschi and Miles [28]. Their work discusseslosses due to flow through holes including inertia but leaves thesolution in the squeeze region with a zero-pressure boundarycondition. However, in the region just above a hole wherethe lateral squeeze flow joins the Poiseuille flow, there is acomplex interaction causing losses that are difficult to estimate.Moreover, the trivial boundary condition (i.e., P = 0) usedin [28] does not hold, and the effect of backpressure due toflow through holes must be accounted for. Darling et al. [30]used arbitrary pressure boundary conditions based on thecomplex acoustical impedance of an aperture and treated thesqueeze-film problem using the Green’s function approach.Kwok et al. [31] used the boundary condition on the holeevaluated for incompressible flow in the numerical simulationperformed using PDEase, which is a finite-element solver. Un-der the assumption of incompressible flow (i.e., a low squeezenumber), the flow through a hole is merely obtained by geomet-rical scaling between the cell diameter and the hole diameter. Ina recent study, Veijola [9] has used existing analytical modelsin the squeeze-film and capillary regions, and using FEMsimulations, he has derived approximate flow resistances at theintermediate region and at the exit from the hole. The modelis valid over a wide range of perforation fractions, from 1% to90%. The said study also presents analysis of cutoff frequenciesfor compressibility and inertia and specifies an upper limit onthe frequency up to which the model is valid.

When the pitch of the holes, the air-gap, and the diameterof the holes are all of the same order, as shown in Fig. 1(a.3),the assumptions made in deriving the Reynolds equation donot hold, and the computationally intensive 3-D Navier–Stokesanalysis becomes indispensable. Moreover, if the outer bordersare open to the ambient, there is a pressure gradient from thecenter of the plate to its borders, and the pressure patternsaround the perforations are not repetitive. In a recent study, asurface extension model has been proposed [32], where the gapsize is comparable to the lateral dimensions. In this approach,it is shown that after extending each open border by (1.3/2)ha,the 2-D Reynolds equation can give accurate results, and the3-D Navier–Stokes equation can be dispensed with. This ap-proach could be applied to the perforated structures as well.

The central motivation of this paper is to provide closed-form expressions to estimate the squeeze-film forces inte-grated with perforation losses and with more realistic boundaryconditions at the holes. When the compressibility and inertialeffects, which are prevalent at high operating frequencies, areincluded in the Reynolds equation, the pressure at the edge ofa hole changes with the frequency of oscillations. The ventingthrough the holes acts as a restrictive passage, and the venting


Fig. 1. (a) Different perforation geometries (ξo is the pitch of the holes, ha is the air-gap height, and l and d are the length and diameter of the holes, respectively).(b) Overview of different squeeze-film models for perforated structures.

boundary condition becomes a complex acoustical impedance.This boundary condition then depends on the hole geometry(i.e., length and diameter) and flow through the hole. Thispressure at the boundary must be evaluated and then appliedas a boundary condition at the edge of the hole near the air-

gap. This apparently circular computation is circumvented bysolving the two flow equations—one for the horizontal flowin the air-gap and the other in the hole—by first assuming anintermediate value of the pressure and then solving for it usingtwo paths of computations discussed in the next section.


Fig. 2. Schematic showing the staggered hole configuration. ξo is the pitch of the holes; ro = 0.525ξo and ri = s/√

π are the area equivalent outer and innerradii of the pressure cell, respectively; rh = s/2 is the equivalent hydrodynamic radius of the hole; ha is the air-gap height; and l is the length of the hole (also thethickness of the back plate). (a) Hexagonal pressure patterns in a staggered configuration. (b) Single equivalent circular pressure cell. (c) Poiseuille flow throughthe perforation.

IV. ANALYTICAL MODEL FOR A CIRCULAR

PRESSURE CELL

A. Modified Reynolds Equation With Inertia, Compressibility,and Gas Rarefaction Effects

In this section, we present the modified Reynolds equationincluding unsteady inertia and compressibility, as well as itssolution, for a single circular pressure cell under the approxi-mation of 1-D radial flow. Isothermal assumptions are used inderiving the model. The derivation follows the general approachgiven by Veijola [2], [3] for rectangular plates separated bya small air-gap. We analytically solve the resulting equation,first for a finite-pressure boundary condition at the edge of thehole. Subsequently, in Section IV-B, we show how to computethis pressure at the edge of the hole by analyzing the flowthrough the hole. A large perforated domain formed by thestaggered hole configuration is shown in Fig. 2(a). This domainis discretized into circular pressure cells, and a single pressurecell considered for the analysis is shown in Fig. 2(b). For thesqueeze-film problem of a single circular pressure cell, let usassume a constant pressure across the gap (∂p/∂z = 0) and nonet tangential flow (∂p/∂θ = 0). The Navier–Stokes momen-tum equation in cylindrical–polar coordinates then reduces to

ρ∂ur

∂t+ ρ

∂ur

∂rur = −∂p

∂r+

∂

∂z

(µ

∂ur

∂z

)(1)

where ur(r, t) is the flow velocity in the radial direction, p(r, t)is the air pressure, and µ and ρ are the viscosity and densityof air, respectively. We assume small air-gap compared to thepitch of the holes and small amplitude harmonic oscillations

(ε = εoejωt). For small air-gap-to-pitch ratios, the lateral veloc-

ity ur is relatively small, and the contribution of the convectiveinertial term ρ(∂ur/∂r)ur can be neglected compared to theunsteady inertial term ρ(∂ur/∂t).

Hence, (1) further reduces to

ρ∂ur

∂t= −∂p

∂r+

∂

∂z

(µ

∂ur

∂z

). (2)

We nondimensionalize (2) using scaled variables Φ=∆p/Pa,Z =z/ha, R=r/ro, τ =ωt, Uz =∂ε/∂τ (i.e., jεoe

jτ ), Ur =ur/Uo, and Uo = roω. Here, Pa is the ambient pressure, ha isthe equilibrium gap thickness, ω is the circular frequency, andro is the outer radius of a pressure cell. The nondimensionalform of (2) becomes

∂2Ur

∂Z2− jRegUr =

1σ

∂Φ∂R

(3)

where σ = µωr2o/Pah2

a, and the modified Reynolds numberReg = ρωh2

a/µ. For very small gaps, or when the pressure islower than the ambient pressure, the no-slip boundary condi-tion may not hold; therefore, we consider the first-order slipboundary conditions on the two oscillating planar structures as

Ur

(R,±1

2

)= ∓Kng

∂Ur

∂Z

(R,±1

2

)(4)

where the nondimensional mean free path Kng = λ/ha is alsothe Knudsen number based on the characteristic length ha andthe mean free path λ. When the pressure is lower than theambient pressure, a quantity Ks = σpKn is used as a measure


of rarefaction, where σp is the slip coefficient [3]. Integrating(3) and applying the boundary conditions (4), we obtain theradial velocity in (5), shown at the bottom of the page.

Integrating (5) w.r.t. Z in the limits (−1/2,+1/2), we getUr(R) averaged in the z-direction as

Ur(R) = − 1Γ

∂Φ∂R

(6)

where Γ = σ/Qpr is a nondimensional complex number thatincludes the compressibility, inertia, and gas rarefaction effects.In the expression for Γ

Qpr =12

−jReg

×[(2 − KngjReg) tan(

√−jReg/2) −

√−jReg

][√

−jReg

(1 − Kng

√−jReg tan(

√−jReg/2)

)]is the relative flow rate coefficient that includes the inertialand gas rarefaction effects based on the first-order slip, andσ is the nondimensional squeeze number accounting for gascompressibility. The central parameter in Γ is the product ofthe Reynolds number and the squeeze number, i.e., Regσ. Thesignificance of this term will be discussed later in Section VI.Next, we derive the modified Reynolds equation in the samemanner as the conventional Reynolds equation [3], [14]. Forthe annular air-gap region, we assume circular symmetry (i.e.,uθ = 0) and integrate the continuity equation in cylindricalcoordinates across the gap from −h/2 to h/2 to get

1r

∂(ρrhur)∂r

+∂(ρh)

∂t= 0 (7)

where ur represents the average velocity such that∫ +h/2

−h/2 urdz = hur.

Furthermore, we nondimensionalize this equation in thesame way as (2) and retain only lower order terms in ε. Usingisothermal flow conditions (p ∝ ρ) and substituting Ur from(6), we get

∂2Φ∂R2

+1R

∂Φ∂R

− Γ(

∂Φ∂τ

+∂ε

∂τ

)= 0. (8)

Making the transformations Ψ = Φ + εo and R =√

jΓR in(8), we get the transformed equation as [33]

∂2Ψ(R)∂R2

+1R

∂Ψ(R)∂R − Ψ(R) = 0. (9)

Equation (9) is in the form of the Bessel equation for animaginary argument of the zeroth order and has a standardsolution for a harmonic motion of the diaphragm given as [33]

Ψ = AI0(R) + BK0(R) (10)

where I0 is the modified Bessel function of the zeroth order,and K0 is the Macdonald’s function of the zeroth order. A andB are complex coefficients that are obtained using the boundaryconditions. Substituting back for Ψ and R, we write the normal-ized complex pressure distribution for the fluid flow governedby (8) over a circular domain for the harmonic velocity of theoscillating plate as

Φ(R, τ) ={AI0(

√jΓR) + BK0(

√jΓR) − εo

}ejτ . (11)

The coefficients A and B in (11) are evaluated by substitutingthe boundary conditions for the pressure cell approximatedwithin a particular pattern. These boundary conditions can betaken as Φ|Ri

= 0 and ∂Φ/∂R|Ro= 0, where Ro and Ri are

the normalized outer and inner radii of the circular pressurecell, respectively. Since the transverse velocity Uz of the plate isassumed to be uniform over the pressure cell, the differentiationof (11) yields

∂Φ∂R

(R, τ) =√

jΓ{AI1(

√jΓR) − BK1(

√jΓR)

}ejτ .

(12)

The zero-radial-gradient boundary condition arises from theassumption of autonomous cells, and the zero-pressure bound-ary condition at the hole is reasonable for low-aspect-ratioholes. This assumption, however, can be relaxed, and a solutioncan be obtained in terms of an arbitrary pressure ΦRi

at thehole. In that case, the constants A and B are evaluated as

A =(ΦRi

+ εo)K1(√

jΓ)[I0(

√jΓRi)K1(

√jΓ) + K0(

√jΓRi)I1(

√jΓ)

]B =

(ΦRi+ εo)I1(

√jΓ)[

I0(√

jΓRi)K1(√

jΓ) + K0(√

jΓRi)I1(√

jΓ)] . (13)

Subsequently, the pressure distribution is obtained in (14),shown at the bottom of the page.

Here, ΦRiis the nondimensional frequency-dependent pres-

sure at the edge of the hole, and its value depends on thebackpressure Φb at the upper end of the hole. In the next section,

Ur(R,Z) =1

jRegσ

(cos(

√−jRegZ)

cos(√

−jReg/2) − Kng

√−jReg sin(

√−jReg/2)

− 1

)∂Φ∂R

(5)

Φ(R) =[(εo + ΦRi

)I0(

√jΓR)K1(

√Γj) + K0(

√jΓR)I1(

√jΓ)

I0(√

jΓRi)K1(√

jΓ) + K0(√

jΓRi)I1(√

jΓ)− εo

]ejτ (14)


Fig. 3. Schematic showing the pressure and velocity boundary conditions in different regions. (a) Horizontal squeeze-film flow. (b) Vertical flow throughthe hole.

we discuss the procedure for obtaining the pressure on the edgeof the hole by considering the loss through the hole.

B. Pressure at the Edge of the Hole

For high-aspect-ratio perforated structures (i.e., l/d > 1),there is a significant loss through the holes that cannot beignored. In addition, this loss will exert a backpressure on thesqueeze-film region, and one cannot apply the trivial pressureboundary condition (p = 0) at the edge of the hole. Thus, itbecomes necessary to find the pressure at the edge of the hole.Analytical models based on extensive FEM simulations aredeveloped for incompressible flow [31]. However, at high fre-quencies, the flow velocity in the squeeze-film region given bythe modified Reynolds equation incorporating compressibilityand inertial effects is frequency dependent. This frequency-dependent flow must be used in the Poiseuille equation to obtaina dynamic pressure. Fig. 3 illustrates the problem with theappropriate boundary conditions for the squeeze-film regionand for the flow through the hole. We model the motion ofthe fluid through the hole as a Poiseuille flow through a pipeincluding inertial effects with atmospheric pressure at the lowerend of the hole.

We start by writing the Navier–Stokes equation for the flowthrough the hole as

ρ∂uzh

∂t= −∂p

∂z+ µ

1r

∂

∂r

(r∂uzh

∂r

)(15)

where uzh is the flow velocity through the hole, and p is thepressure along the axis of the hole. The flow through the holes

oscillates with the same frequency ω as that of the plate. Byapplying the ambient pressure boundary condition on the freeend of the hole, we write (15) as

1r

∂

∂r

(r∂uzh

∂r

)+ α2uzh = −p

b

µl(16)

where pb

represents the pressure on the upper edge of thehole, and α =

√(−jωρ)/µ is the complex frequency vari-

able (αrh =√−jReh, and Reh = ρωr2

h/µ is the modifiedReynolds number for the flow through the holes). We solve thisequation with ∂uzh/∂r = 0 at r = 0, and the first-order slipboundary condition uzh = −λ(∂uzh)/∂r at r = rh, to get

uzh = − pb

jωρl

(1 − J0(αr)

J0(αrh) − λαJ1(αrh)

). (17)

Veijola [9] has extracted entry and exit losses by performingextensive FEM simulations and presented loss factors in theform of relative elongations. We have not used the relativeelongation at the exit as the ambient pressure is applied rightat the outlet of the hole in the numerical simulations used forvalidation. The relative elongation corresponding to the entryloss is used in this analysis, and the effective length of a hole isobtained as leff = (l + ∆c ri), where ∆c is given by

∆c = (1 + 0.6 Knh)(

0.66 − 0.41ri

ro− 0.25

r2i

r2o

). (18)

By integrating the flow velocity given by (17), we ob-tain the total volume rate of flow through the hole as


Vzh =∫ rh

0 uzh2πrdr. Thus, we get

pb = − jωρleff(1 − 2

αrh

J1(αrh)J0(αrh)−λαJ1(αrh)

) Vzh

πr2h

. (19)

We divide pb by the atmospheric pressure Pa to get thenondimensional pressure Φb on the edge of the hole. Also,we obtain the volume flow rate (Vzh = Vr + Vz) through thehole by summing the volume flow rate (Vr = 2πrhhaur) in theradial direction evaluated at rh and the axial volume flow fromabove the hole (Vz = πr2

huz). These substitutions give

Φb = δ1

(uz +

2ha

rhur

)

where

δ1 =−jωρleff

Pa

(1 − 2

αrh

J1(αrh)J0(αrh)−λαJ1(αrh)

) . (20)

Substituting Ur from (6) in ur = roωUr, we get

ur = −roω

Γ∂Φ∂R

. (21)

We first substitute the pressure gradient ∂Φ/∂R|R=rigiven

by (14), evaluated at the edge of the hole in (21), to get

ur|ri(R) = −jroω√

jΓ

{AI1(

√jΓRi)−BK1(

√jΓRi)

}.

(22)Next, we substitute the coefficients A and B to get

ur|ri= jroω(ΦRi

+ εo)δ2

where

δ2 =− 1√jΓ

[K1(

√jΓ)I1(

√jΓRi) − I1(

√jΓ)K1(

√jΓRi)

I0(√

jΓRi)K1(√

jΓ) + (K0

√jΓRi)I1(

√jΓ)

].

(23)

Finally, we substitute for ur|riin (20) and obtain the

frequency-dependent pressure boundary condition on the edgeof the hole as

Φb = δ1

[uz +

2ha

rijroω(ΦRi

+ εo)δ2

]. (24)

In (24), we can substitute ΦRi= Φb, which turns out to be

a reasonable assumption based on the values of pressure gradi-ents in the radial and z-directions observed in our simulations.We find that the radial pressure gradient is at least one orderof magnitude smaller than the vertical gradient, making the

relative pressure variation in the radial direction very small. Infact, the pressure difference between the edge of the hole andthe center at the hole opening is between 2% and 4% in allsimulated cases. Now, solving for Φb, we get

Φb =δ1(1 + 2ro

riδ2)

1 − j 2haroωri

δ1δ2

uz. (25)

The parameter δ1 depends on the perforation geometry andtakes into account the frequency-dependent inertial effectsbased on the first-order slip flow through the holes. The secondparameter δ2 depends on the geometry of the annular squeeze-film region and also takes into account the frequency-dependentcompressibility and inertial effects based on the first-order slipflow through the air gap. Substituting ΦRi

= Φb in (14), we getthe normalized complex force offered by the pressure cell in theannular region as

Fsq = 2π

Ro∫Ri

Φ(R)RdR. (26)

Evaluating the above integration over the annular region, weget the complex force due to the squeeze-film Fsq = Fsq1 +Fsq2 given by (27) and (28), shown at the bottom of the page.

Equations (27) and (28) give the squeeze-film force in theannular region of the air-gap due to the small harmonic os-cillations of the plate and due to the backpressure caused bythe flow through the hole, respectively. Note that for a thinperforated plate and large holes (more venting), the pressuredrop through the hole Φb is negligible, and the boundary con-dition approaches the ambient pressure (as one would expect).In that case, the contribution of (28) becomes negligible, and(27) alone can accurately predict squeeze-film damping. Next,the force acting just above the hole is obtained as

Fh =[(

πR2i

)Φb

]ejτ . (29)

The net force on a pressure cell is obtained by adding theforces given by (27)–(29) as

Fnet = Fsq1 + Fsq2 + Fh. (30)

Finally, we have the damping force and the combined stiff-ness and inertia force offered by the pressure cell as

Fdamp. = {�(Fnet)}Par2o

Fstiff.+inr. = {�(Fnet)}Par2o (31)

Fsq1 =πεo

[2Ri√

jΓ

[I1(

√jΓ)K1(

√jΓRi) − I1(

√jΓRi)K1(

√jΓ)

][I0(

√jΓRi)K1(

√jΓ) + K0(

√jΓRi)I1(

√jΓ)

] −(1 − R2

i

)]ejτ (27)

Fsq2 =πΦb

[2Ri√

jΓ

[I1(

√jΓ)K1(

√jΓRi) − I1(

√jΓRi)K1(

√jΓ)

][I0(

√jΓRi)K1(

√jΓ) + K0(

√jΓRi)I1(

√jΓ)

]]

ejτ (28)


TABLE IEQUIVALENCE BETWEEN GEOMETRIC VARIABLES OF

ANALYTICAL AND NUMERICAL MODELS

where � and � represent the imaginary and real parts of theircomplex arguments, respectively. The phase angle between theimparted velocity and the force response is given as

θ = arctan�(Fnet)�(Fnet)

. (32)

The efficacy of this analytical model is assessed by com-paring the analytical results with the numerical simulationsusing the 3-D Navier–Stokes solver ANSYS-CFX. In the nextsection, we briefly discuss the numerical model developed inANSYS-CFX.

V. NUMERICAL MODEL AND SIMULATIONS

Generally, perforations are present in staggered and non-staggered (matrix) configurations. This gives rise to a hexag-onal squeeze-film region in the staggered hole configurationand a square squeeze-film region in the nonstaggered holeconfiguration (in the x−y plane). In addition, square etchhole geometries are very common (in the z-direction). In theanalytical model, the squeeze-film region and the perforationsare assumed to be circular with equivalent radii, as shownin Table I. This simplifies the analytical procedure and givesclosed-form solutions. However, in the numerical simulationsperformed using ANSYS-CFX, the fluid cells are modeled withthe actual geometry (i.e., a hexagonal or a square cell with asquare hole).

ANSYS-CFX makes use of the finite volume method. Fig. 4illustrates a typical fluid volume modeled using half-symmetry.The entire fluid domain is discretized using a hexahedral meshcomprising 15 000 to 30 000 elements for different geometries.A fine mesh is used in the intermediate region above the holeand adjacent to the solid walls. A mesh density of more than15 elements/µm3 ensures mesh independent analysis, giving aconverged solution. The pressure symmetry boundary condition(∂p/∂n = 0) is applied on the hexagonal periphery.

Losses at the exit of the hole are neglected, and the ambientpressure (i.e., zero relative pressure, p = 0) is applied at the freeend of the hole. The flow is assumed to be isothermal, and slip

Fig. 4. Hexagonal cell with a square hole (only half-volume) meshed usinghexahedral elements for simulations in ANSYS-CFX for ha = 2 µm, s =4 µm, and l = 10 µm.

Fig. 5. Steady-state force response to a harmonic velocity applied to the fluidsurface in the hexagonal cell simulated in ANSYS-CFX.

boundary conditions are used on the solid walls. A harmonicvelocity with a small amplitude of oscillations (εo = 0.075 ×ha), comprising 80 time steps over one cycle of oscillation, isapplied normal to the top fluid surface, and transient analysisis performed in each case. We simulated the system responseuntil it reached a steady state. To ensure that the steady statewas indeed reached, we computed the response for a few cyclesand compared the phase in the current cycle with that in theprevious cycle by computing (θn − θn−1)/θn and restrictingthis value to less than 1% to declare convergence. We found thatbecause of the fine time discretization used, convergence wasreached within two to ten cycles for all frequencies, with higherfrequencies requiring a larger number of cycles. The averagepressure acting on the oscillating plate is then extracted, andthe net force is obtained by multiplying the average pressure bythe area of the plate. Fig. 5 shows a steady-state force responseto the imparted harmonic velocity at 1-MHz frequency for acell with a 2-µm air gap and an 8-µm-wide × 10-µm-deepperforation. The phase shift between the velocity and the forceis calculated as θ = 2πf∆t, where ∆t is the time lag or leadbetween the velocity and the force, and f is the frequency ofoscillations. The pressure distributions for the case of hexag-onal and square perforated cells are shown in Fig. 6. Theprimary focus of this study is to examine the behavior ofthe analytical model that includes compressibility and inertialeffects of the fluid and takes into account the loss through theholes. A total of about 500 numerical simulations are carried


Fig. 6. Pressure profiles in perforated hexagonal and square cells simulated in the 3-D Navier–Stokes solver ANSYS-CFX.

TABLE IIDIMENSIONS AND SIGNIFICANT GEOMETRIC RATIOS OF THE PERFORATED

CELLS MODELED FOR SIMULATIONS IN ANSYS-CFX

out on 36 different geometries of hexagonal cells. For eachgeometry, 10–15 frequency points between 5 kHz and 50 MHzare simulated. The geometries and simulation parameters aresummarized in Table II.

The analytical model is based on the frequency-domain formof equations that directly give the frequency response (ampli-tude of the force), whereas the numerical results are obtainedby performing transient analysis at discrete frequencies. Inthe following sections, we examine the performance of theanalytical model by comparing the analytical results with thenumerical results over a wide range of frequencies. The resultsof the comparison are followed by a brief discussion.

VI. RESULTS AND DISCUSSION

A. Inertial Effects and Loss Through the Hole

In our previous work [29], we presented a 2-D analyticalsqueeze-film damping model incorporating compressibility. Inthat model, the losses due to the flow through the holes wereneglected, and the ambient pressure boundary condition wasused on the hole. Moreover, the inertial effects were not mod-eled. These effects can be significant at high frequencies and forlarge air-gap heights. Thus, the model was applicable for rela-tively larger perforations and for low Reynolds number regime

Fig. 7. Frequency response (amplitude) of the squeeze-film model in compar-ison with the results without inertia and loss through the holes (i.e., Φb = 0)and the numerically simulated model in ANSYS-CFX (×) for ξo = 20 µm,ha = 4 µm, s = 8 µm (perforation ratio of 18.5%), and l = 20 µm.

(Re < 1). The motivation for this work is derived from the needto overcome these limitations and develop an improved model.Therefore, we first examine the response of this model, whichincludes the loss through the hole and inertial effects. For thispurpose, we choose a perforation geometry having a 4-µm air-gap (ha), an 8-µm hole size (s), and a 20-µm length of thehole (l). Fig. 7 shows the frequency response results obtainedusing the analytical model under different assumptions. First,the ambient pressure boundary condition Φb = 0 is set onthe upper edge of the hole (i.e., the loss through the hole isneglected), and inertial effects are also suppressed by settingthe flow rate coefficient Qpr = 1 (i.e., setting Γ = σ) in the ex-pression for forces. The frequency response resembling that ofa first-order system, consisting of a spring and a damper, shownby a dash-dot line is obtained in this case. Next, the inertialeffects in the air-gap are included (i.e., by resetting Γ), but theambient pressure boundary condition Φb = 0 is still retained.In this case, the inertial effect grows with the frequency of


Fig. 8. Comparison of the net force (solid line) and damping force (dashed line) obtained using analytical formulas with those obtained from ANSYS-CFX(× indicates net force, and ◦ indicates damping force) for fixed perforation pitch, ξo = 20 µm, but varying hole geometry and air gap. (a) Size of the holes = 8 µm (corresponding perforation ratio of 18.5%), ha = 2 and 4 µm, and l = 1, 10, and 20 µm. (b) Size of the hole s = 10 µm (corresponding perforationratio of 29%), ha = 2 and 4 µm, and l = 1, 10, and 20 µm.

oscillations and so does the compressibility effect. The resultsindicated by a dotted line show a resonance peak, implyingthat the system behaves as a second-order spring–mass–dampersystem. Finally, the ambient pressure boundary condition isapplied on the lower edge of the hole; thereby, Φb gets a finitevalue corresponding to the loss through the hole, and results ofa full 3-D flow are obtained, as shown by the solid line. Theresults obtained using the numerical model are also plotted forcomparison. The offset in the frequency response indicated in

Fig. 7 corresponds to the loss through the hole. The onset ofinertial effects is seen from 100 kHz onward, with the solidand dotted lines gradually deviating upward, leading to the firstresonance.

B. Comparison of Analytical and Numerical Results

Next, we compare the behavior of the analytical model withthe numerical model for different perforation ratios, air-gap


TABLE IIIRELATIVE ERRORS IN ANALYTICAL RESONANCE FREQUENCIES, DAMPING FORCE, AND NET FORCE w.r.t. THE NUMERICAL RESULTS FOR VARIOUS

GEOMETRIES. THE ERROR IS ESTIMATED OVER TWO FREQUENCY REGIMES: L INDICATES THE LINEAR AND MODERATELY NONLINEAR REGION

BEFORE THE RESONANCE (f < 0.8fn), AND NL INDICATES THE STRONGLY NONLINEAR REGION

BEYOND THE RESONANCE (f > 0.8fn), AROUND THE RESONANCE PEAK

heights, and forcing frequencies. A perforated cell of pitchξo = 20 µm is selected. Perforation sizes of s = 8 µm ands = 10 µm give the perforation ratios of 18.5% and 29%, re-spectively. These, along with two air-gap heights ha (i.e., 2 and4 µm) and three perforation lengths l (i.e., 1, 10, and 20 µm)give 12 combinations. The frequency response curves areshown in Fig. 8(a) and (b). The frequency response amplitude(i.e., the net force) and the damping force due to the squeezefilm are very accurately predicted up to a frequency f = 0.8fn

(where fn is the first resonance frequency) with less than 20%error for long hole lengths (l = 10 and 20 µm) and less than30% error for short hole lengths (l = 1). The damping forcereaches its peak at the resonance and rapidly decreases beyondthe resonance, showing a good agreement with the numericallyobtained damping force. However, beyond 0.8fn, the frequencyresponse amplitude is overestimated using the analytical model,and in most cases, the error is as high as 80%. This could bedue to the fact that the analytical model neglects the convectiveinertial term. Resonance frequencies predicted by the analyticalmodel and those obtained by interpolating the numerical resultsare listed in Table III, along with the percentage error. Theresonance frequency decreases with the increase in the lengthof the holes, which is expected (Fig. 8(a) and (b), left to right)due to the increased mass of air in the long holes. Furthermore,when the air-gap increases from 2 to 4 µm, compressibilitydrops, and the mass of air increases, lowering the resonancefrequencies.

In Fig. 9, we study the effect of perforation size s on thenature of the frequency response. In this case, the perforationlength is kept constant at 20 µm. Four air-gap heights (i.e., 0.5,1, 2, and 4 µm) and three perforation sizes (i.e., 4, 8, and 10 µm)are considered, giving 12 different geometries. In the case of4-µm perforations, compressibility predominates, and no visi-ble resonance peak is seen for all the air-gap heights. However,in the case of 8- and 10-µm perforation sizes, the resonance

peak is seen for all air-gap values except for the 0.5-µm air-gap. As the frequency of oscillations enters the megahertzregime, the pressure response obtained analytically lies abovethe response obtained numerically. It should be noted that thenonlinear inertial term ur(∂ur/∂r) is neglected while derivinganalytical equations for both the squeeze-film and pipe flows.In particular, beyond the resonance frequency, the contributionof the nonlinear inertial term may not be negligible. The effectis even more pronounced for higher perforation lengths dueto the fact that the analytical inertial effect through the pipeis underestimated in the analytical formulas. At even higherfrequencies, i.e., beyond 20 MHz, the disagreement between theanalytical and numerical results is even more, and sometimes,the numerical results drastically fluctuate. At velocities corre-sponding to these frequencies, one should expect other unmod-eled effects such as lateral resonances in the air-gap, etc. Theonset of such effects can be detected by a term obtained by theproduct of the Reynolds number and the squeeze number, i.e.,Regσ = 12((roω)2/c2) (substituting Pa = ρc2, where c is theisothermal sound speed). Since roω is used to normalize theradial velocity ur, this ratio is directly useful in predictingthe radial flow velocity getting close to acoustically complexphenomena. The term Regσ is independent of the air-gapheight and varies with the square of the oscillation frequency.We find that around 20 MHz, the comparison between theanalytical and numerical models breaks down. At 10-MHzfrequency, the ratio (roω)/c is close to 2. In actuality, at suchhigh frequencies, the acoustic wavelengths become comparableto the pitch of the pressure cell and the perforation length.The acoustic effects in the lateral direction and in the holeare likely to show up at these frequencies. These effects arenot captured by the analytical model. We also point out thatat such frequencies, the isothermal flow assumption becomesquestionable. At 20 MHz, the time period of oscillation is 50 ns,and the time taken for heat propagation from the center of a


Fig. 9. Comparison of the frequency response curves obtained using analytical formulas and ANSYS-CFX (◦,×, ∗) for different air-gap heights and the size ofthe holes (the pitch of the holes ξo = 20 µm, and the length of the holes l = 20 µm).

2-µm-thick squeeze film to the solid structural surface (i.e.,to travel 1 µm) is about 15 ns (calculated using the velocityof heat propagation in air, i.e., v =

√4πfκ, where f is the

frequency, and κ is the diffusivity of air) [34]. Thus, the twotimescales are comparable, and the isothermal flow assumptionis no longer valid. Therefore, although the analytical resultsare in good agreement with the numerical results from theANSYS-CFX simulations until 20-MHz frequency, we recom-mend using the analytical model only with caution beyond thefirst resonance peak.

C. Comparison of Compressibility and Inertial Effects in a2-D Model

In Fig. 10, we make a comparison of the frequency re-sponse curves obtained using the analytical formulas for fourdifferent air-gap heights (i.e., 0.5, 1, 2, and 4 µm). For lowfrequencies, the response amplitude linearly rises with thefrequency of oscillation. This is obviously due to the fact thatat low frequencies, the flow is predominantly viscous, whichis easily identifiable because both the Reynolds number Reg

and the squeeze number σ are low (� 1). With the increasein frequency, the inertia and compressibility of air in the air-gap continuously build up, and for different values of the air-gap height, the nature of the response curve considerably varies

Fig. 10. Comparison of the analytically obtained frequency responses of a2-D model (i.e., Φb = 0) for different air-gap heights (the pitch of the holesξo = 20 µm, and the size of the hole s = 8 µm). Two points are marked oneach response curve, where σ = 1 and Reg = 1. Different shades of gray areused to delineate regions where one or the other effect has a significant effecton force calculations.


with frequency. The contribution of compressibility and inertiadepends on the geometry of the cell and the frequency ofoscillations. Since Reg and σ linearly vary with frequency, thequestion is which number rises faster with frequency. This ispurely a function of geometry. For a given pitch of the holesand the hole diameter, air-gap ha is the most critical parameteraffecting inertia and compressibility. As the air-gap heightincreases, the inertial contribution, and hence the Reynoldsnumber (∝ h2

a), starts to increase, whereas the compressibilityeffect (∝ 1/h2

a) starts to decrease. We propose to use theratio of the Reynolds number to the squeeze number as acharacteristic number to capture the relative effect of inertiaand compressibility. Let γ = (Reg/σ). For hole size s = 8 µmand pitch of the holes ξo = 20 µm, γ = 1 for ha = 1.34 µm.When γ < 1, compressibility dominates. On the other hand,if γ > 1, Reg rises faster, which means that inertial effectsdominate throughout over compressibility. This is clearly seenfrom the set of response curves for four different air-gap heightsin Fig. 10. For a given geometry, the Reynolds number and thesqueeze number are indicative of the frequency at which theinertial and compressibility effects start to dominate overthe viscous effects. Since the ratio γ eliminates frequency,the effect of geometry is well brought out by this ratio. It is ob-served that γ varies as the fourth power of ha, and hence, fromha = 0.5 to 4µm, the ratio goes up by more than three ordersof magnitude, switching the behavior from predominantly com-pressible to mildly compressible and then inertial. This shouldexplain the rapid change from a first-order to a second-orderfrequency response in Fig. 10. The shaded regions indicatethe regimes dominated by the damping, compressibility, andinertial effects.

VII. CONCLUSION

A compact squeeze-film model was developed for a perfo-rated cell that includes compressibility, inertia, and gas rarefac-tion effects. The analytical results obtained were compared withthe numerical simulations carried out using the finite volumemethod in ANSYS-CFX. The analytical frequency responseand the damping force up to a frequency f = 0.8fn (where fn

is the first resonance frequency) match within 20% accuracyfor high-aspect-ratio perforations and 30% accuracy for low-aspect-ratio perforations. A large error for thinner perforations(smaller hole lengths) suggests that the treatment of orifice-like flow with pipe flow (as done here) requires a furtherrefinement to handle this limiting case. The first resonancewas predicted within 21% error depending on the length ofthe holes. The results clearly show that with the inclusion ofinertia and compressibility, the analytical model does very wellin predicting the response. However, this model is inadequatefor response prediction beyond f = 0.8fn (i.e., around the firstresonance frequency) because of the inherent limitations of thefluid flow assumptions made in the derivation of the governingequations. At very high frequencies, the pitch and length of theholes become comparable to the acoustic wavelengths, invitingcomplex interactions that are not accounted for in this model.Finally, it should be noted that the model presented here is bestsuited for applications with closed air-gap borders.

ACKNOWLEDGMENT

The authors would like to thank R. B. Yarasu, CGP Labora-tory, for his help in carrying out the Navier–Stokes simulationsin ANSYS-CFX and Prof. G. Fedder, Associate Editor, forsuggesting a simulation check that led to an improved modelfor the squeeze-film force calculations. Thanks are also due toA. K. Pandey for valuable inputs.

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Suhas S. Mohite received the Bachelor’s degree inmechanical engineering from the Walchand Collegeof Engineering, Sangli, India, in 1988, the Master’sdegree in machine tools from the Indian Instituteof Technology, Madras, India, in 1990, and thePh.D. degree in microelectromechanical systemsfrom the Indian Institute of Science, Bangalore,India, in 2008.

He was with Bajaj Auto Ltd., Pune, India, andWipro Fluid Power Ltd., Bangalore. He is currentlya Professor with the Department of Mechanical

Engineering, Government College of Engineering, Karad, India. His researchinterests include dynamics and control, condition monitoring, computer-aided design and manufacturing, mechatronics, and design and modeling ofMEMS sensors.

Venkata R. Sonti received the B.Tech. degreefrom the Indian Institute of Technology, Kharagpur,India, in 1986, and the Ph.D. degree in mechanicalengineering from Purdue University, West Lafayette,IN, specializing in digital signal processing, controls,acoustics, and vibration.

Prior to joining the Indian Institute of Science,Bangalore, India, in 1999, he was with Caterpillar,USA, as a Specialist Engineer. He is currently anAssistant Professor with the Department of Mechan-ical Engineering, Indian Institute of Science. He has

worked in active noise control for several years and holds practical skillsin product development and testing. His research interests include structuraldynamics (vibrations of plates and shells), fluid-structure interaction, acoustics,modeling joints in built-up structures, underwater noise from turbulent flows,development of MEMS acoustic sensors, and sensors for active control ofbroadband noise.

Rudra Pratap (M’07) received the B.Tech. de-gree from the Indian Institute of Technology (IIT),Kharagpur, India, in 1985, the M.S. degree in me-chanics from the University of Arizona, Tucson, in1987, and the Ph.D. degree in theoretical and ap-plied mechanics from Cornell University, Ithaca, NY,in 1993.

From 1993 to 1996, he was with the Sibley Schoolof Mechanical and Aerospace Engineering, CornellUniversity. In 1996, he was with the Indian Instituteof Science, Bangalore, India. From 1997 to 2000, he

was an Adjunct Assistant Professor with Cornell University. He is currently anAssociate Professor with the Department of Mechanical Engineering, Indian In-stitute of Science. His research interests include MEMS design, computationalmechanics, nonlinear dynamics, structural vibration, and vibroacoustics.

Dr. Pratap is a member of the Institute of Smart Structures and Systems. Heis on the Editorial Board of the journal Computers, Materials, and Continua.

a compact squeeze-film model including inertia ...eprints.iisc.ac.in/14924/1/fulltext.pdf · a...

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