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Precision Engineering 36 (2012) 399– 407

Contents lists available at SciVerse ScienceDirect

Precision Engineering

jou rna l h om epage: www.elsev ier .com/ locate /prec is ion

erformance of inherently compensated flat pad aerostatic bearings subject toynamic perturbation forces

ikhil Bhata,∗, Senthil Kumarb, Wayne Tanc, Ramarthinam Narasimhand, Tsu Chuin Lowe

Center of Innovation, Ngee Ann Polytechnic, SingaporeDepartment of Mechanical Engineering, National University of Singapore, SingaporeDSO National Laboratories, SingaporeDepartment of Mechanical Engineering, Indian Institute of Science, Bangalore, IndiaMakino Asia Pte Ltd., Singapore

r t i c l e i n f o

rticle history:eceived 5 May 2011eceived in revised form 1 November 2011ccepted 5 January 2012vailable online 21 January 2012

eywords:ir bearingtatic and dynamic performancenherent compensation

a b s t r a c t

The importance of air bearing design is growing in engineering. As the trend to precision and ultraprecision manufacture gains pace and the drive to higher quality and more reliable products continues,the advantages which can be gained from applying aerostatic bearings to machine tools, instrumentationand test rigs is becoming more apparent. The inlet restrictor design is significant for air bearings because itaffects the static and dynamic performance of the air bearing. For instance pocketed orifice bearings givehigher load capacity as compared to inherently compensated orifice type bearings, however inherentlycompensated orifices, also known as laminar flow restrictors are known to give highly stable air bearingsystems (less prone to pneumatic hammer) as compared to pocketed orifice air bearing systems. However,they are not commonly used because of the difficulties encountered in manufacturing and assemblyof the orifice designs. This paper aims to analyse the static and dynamic characteristics of inherentlycompensated orifice based flat pad air bearing system. Based on Reynolds equation and mass conservation

equation for incompressible flow, the steady state characteristics are studied while the dynamic statecharacteristics are performed in a similar manner however, using the above equations for compressibleflow. Steady state experiments were also performed for a single orifice air bearing and the results arecompared to that obtained from theoretical studies. A technique to ease the assembly of orifices with theair bearing plate has also been discussed so as to make the manufacturing of the inherently compensatedbearings more commercially viable.

. Introduction

Air bearings have been gaining more attention recently as theyffer virtually frictionless motion due to low viscosity of fluid (air).t the same time, air bearings consume less power as compared toil lubricated bearings [1]. This is beneficial especially consideringhe current trend of machining, i.e. towards precision, micro- orano-machining. Air bearings with improved static and dynamicerformance characteristics can help improve the performance ofrecision machining.

In terms of air bearings, they can be categorized based on shapeplain bearing and journal bearing) [1] or pressure profile gen-ration (hydrostatic, hydrodynamic and hybrid bearing) [2]. For

∗ Corresponding author.E-mail addresses: bnj2@np.edu.sg (N. Bhat), mpeask@nus.edu.sg (S. Kumar),

nyawnat@gmail.com (W. Tan), narasi@mecheng.iisc.ernet.in (R. Narasimhan),owtc@makino.com.sg (T.C. Low).

141-6359/$ – see front matter © 2012 Elsevier Inc. All rights reserved.oi:10.1016/j.precisioneng.2012.01.002

© 2012 Elsevier Inc. All rights reserved.

this research, inherently compensated plain hydrostatic bearing (ora.k.a. thrust bearings) is chosen for its translation function.

Hydrostatic bearing is investigated for its stability to performwith or without relative motion between the bearing plates. For therestrictors, inherent compensated orifice will be used since it pro-vides more stable (due to laminar flow characteristics) performanceas compared to pocketed orifice (which suffer from pneumatichammer problem) and porous restrictor (which can easily clog dueto porous nature of the restrictor). Also the recent advances in nee-dle tubing popularly called as hypodermic needle tubing, can makethe manufacturing of inherently compensated bearings consider-ably easy as compared to pocketed or porous bearing. This will bediscussed in Section 5.

A lot of research has been done with regards to modelling theperformance of an air bearing to study its steady and dynamic

state characteristics. For hydrostatic air bearings, many researchstudies were reported on circular flat pad bearings. Differentapproaches for evaluating the steady state performance such asanalytical techniques, CFD simulation, numerical iteration methods

400 N. Bhat et al. / Precision Enginee

Nomenclature

List of variablesA area of air bearing surface (m2)

|Be|

∂Nei

∂x0

∂Nej

∂x0 · · ·

0∂Ne

i

∂x0

∂Nej

∂x· · ·

∂Nei

∂x0

∂Nej

∂x0 · · ·

0∂Ne

i

∂x0

∂Nej

∂x· · ·

C coefficient of damping

|C|∣∣∣∣ −c(x, y) 0

0 c(x, y)

∣∣∣∣d capillary/orifice diameter (m)d(x, y) d = 12�ωh1

h30

p0

h bearing gap height (m)f1 real part of Z1f1(x, y) approximate solution of f1|F| force matrixg1 imaginary part of Z1g1(x, y) approximate solution of g1h0 steady-state bearing gap height (m)h1 dynamic-state (perturbated) bearing gap height (m)i imaginary rootK coefficient of stiffness|K| stiffness matrixLC length of capillary/orifice (m)min mass flow rate of air into orifice (kg/s)mout mass flow rate of air into bearing gap (kg/s)mx mass flow rate per unit length in x-direction

|Ne|∣∣∣∣ Ne

i(x, y) 0 Ne

j(x, y) 0 · · ·

0 Nei

0 Nej

· · ·

∣∣∣∣|Ne|

∣∣∣∣ 0 Nei

0 Nej

· · ·Ne

i0 Ne

j0 · · ·

∣∣∣∣P air pressure (Pa)PS supply air pressure (Pa)PD pressure entering air gap (Pa)P0 steady-state air pressure within bearing gap (Pa)P1 dynamic-state (perturbated) air pressure within

bearing gap (Pa)R gas constant (287 J/kg K)T temperature (K)t time|U| matrix of f1 and g1 termsx displacement in x-axisy displacement in y-axisw load capacity of air bearing (N)w1 dynamic load capacity of air bearing (N)W1 weighting function (Eq. (13))W2 weighting function (Eq. (13))� density of air (kg/m3)� kinetic viscosity of air (m2/s)ω frequency (rad/s)�() gradient operator, ∇( ) = ∂ ( ) + ∂ ( ) + ∂ ( )

asoo[

∂x ∂y ∂z

� pi

nd experiments were used for pocketed orifice and gradient profile

urface circular flat pad bearings [3–5]. Similar studies were carriedut to evaluate the dynamic characteristics of bearing dependingn the type of orifice restrictor and air bearing operating conditions6–10].

ring 36 (2012) 399– 407

The steady performance of a pocketed orifice air bearings fordifferent operating condition was modelled in Fluent simulationsoftware and experimental studies were carried out to verify theresults [11,12]. Mishra [13] modelled the dynamic instability oftilted rectangular flat pad bearing using analytical method. Stoutand Sweeney [14,15] formulated design charts for rectangular flatpad bearings with both inherent and pocketed orifice restrictors.This is to enable quick selection and manufacture of bearings forindustrial usage.

To improve the dynamic performance characteristics of porousaerostatic bearings a surface restricted layer was formed on the sur-face and the influence of the surface restricted layer on the dynamicstiffness and damping coefficient was studied [16]. It was provedboth numerically and experimentally that the porous metal bear-ings with relatively large permeability and small surface restrictionratio can achieve a higher dynamic stiffness and a larger damp-ing coefficient. Miyatake and Yoshimoto [17] studied the static anddynamic performance of annular circular thrust bearings with smallfeed holes (<0.05 mm). The analysis was done numerically usingCFD. It was found that aerostatic bearings with small feed holeshave stiffness similar to the bearings with compound restrictorsand a higher damping coefficient. Mukai [18] described a methodfor calculating the damping characteristics of an air bearing basedon fluid dynamics and the structure of a small gap air bearing thatexhibits an excellent vibration damping characteristics.

Chen et al. [19] evaluated the stiffness of various geometricdesigns of aerostatic journal bearings for high-speed spindles underdifferent operating conditions. First, the stiffness of the front andrear journal bearings was evaluated experimentally using the rela-tionship of force and displacement at different supply pressures. Anumerical model was then developed to simulate journal-bearingstiffness under the same pressure range to validate the results. Themodel was extended to calculate the stiffness of the designs withvarious geometric parameters such as L/D ratio, different typesof restrictor designs (e.g., pocketed orifice versus inherent ori-fice); supply orifice diameter; radius gap; and supply pressures.It was found that gas-bearing geometries had a significant effecton stiffness. The results also provide helpful design guidelines forgas-bearing spindles in high-precision machine applications.

Tala-Ighil et al. [20] examined the texture location influenceon the hydrodynamic journal bearing performance. A numericalmodelling is used to analyse the cylindrical texture shape effecton the characteristics of a hydrodynamic journal bearing. The the-oretical results showed that the most important characteristicssuch as the minimum film thickness and friction torque can beimproved through an appropriate arrangement of the textured areaon the contact surface. Li and Ding [21] developed a high accelera-tion and high precision xy stage using aerostatic linear guidewaysand manufactured it to meet the requirements of chip manufac-turing equipment. The finite-element method used to calculatethe performance of the linear guideways. The design parameterssuch as orifice diameter, distribution of orifices, and film thick-ness are determined by calculating the load carrying capacity, tiltstiffness, and mass flow-rate of the guideways. Aguirre et al. [22]proposed a new multiphysics finite element model which consid-ers the interaction between the air flow dynamics, the structuralflexibility of the bearing, the piezoelectricity for the actuators andthe control with strongly coupled formulation. The main aim ofthis research was to pave development of an aerostatic slide withactive bearings, aiming at nanometre position control in all degreesof freedom. However the outstanding expected performance comesat the expense of design complexity, due to the amount of relevant

design parameters belonging to different physical fields. Recentlythere has been an active interest in air bearings combined withmagnetic preloads [23]. The magnetic actuators for preloadingair bearings are combined with permanent magnets and coils to

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N. Bhat et al. / Precision E

djust the air bearing clearance by actively controlling the mag-etic force. This helps in the compensation of motion errors to giveltra-precise straight motion. Although these magnetically pre-

oaded air bearings provide a simpler configuration as comparedo conventional wrap-around air bearing designs, their dynamicerformance (subject to perturbation forces) has not been studied

n detail.There has also been a growing interest in texturing hydrody-

amic bearings [20] to improve the performance of air bearings.owever texturing appears to be least effective in improving the

oad capacity of the bearing and a careful consideration needs to beiven for appropriate location of the textured area. Another methodo improve the performance of aerostatic bearings has been dis-ussed by Bhat et al. [24] wherein form errors are induced on theurface of air bearings to improve the performance of aerostaticearings mainly in terms of stiffness. However careful consider-tion has to be given to the magnitude of induced form error.hat et al. [25] investigated the dynamic performance of inherentlyompensated aerostatic flat pad air bearings. The effect of param-ters like orifice diameter, gap height and supply pressure on theynamic performance of the air bearing was studied.

In conclusion, not much research work has been done on theectangular inherently compensated hydrostatic air bearings espe-ially with regards to evaluating their dynamic characteristics.ectangular shaped air bearing are useful in the design of machineool as it can be guided easily due to geometrical symmetry.ence, this research will concentrate on modelling the static andynamic performance characteristics of the inherently compen-ated rectangular flat pad bearings and extend the work done in25]. Experimental techniques to validate the FEM model will alsoe discussed. Also, a technique to ease the assembly of orificesith the air bearing plate will also be discussed so as to make theanufacturing of the inherently compensated bearings more com-ercially viable. Finally recommendations are made to deign high

erformance aerostatic bearings with good dynamic stability.

. Model design and equations

The steady and dynamic performance characteristics will benvestigated. In order to study these, the pressure profile must bebtained through modelling of the flow of air through the bearing.he main parameters of analysis are the coefficient of stiffness andoefficient of damping. These will give a better understanding ofhe stability of the air bearing. Another important parameter to beonsidered as well is the load capacity of the bearing.

.1. Air bearing model

The bearing, as shown in Fig. 1, forms an air film in between

he bearing surface and the reaction plate. This air film is beingupplied by an external pressurized air source (1) and it goeshrough the bearing and exits the orifice at the bearing surface2). The pressure profile can be described as the orifice having the

Fig. 1. Air bearing model.

ring 36 (2012) 399– 407 401

highest pressure (3) and the pressure reduces as it moves furtheraway until it reaches atmospheric pressure at the boundary (4).Therefore modelling the air flow is very important so as to under-stand the pressure distribution within the gap.

2.2. Basic equations

The air passing through the bearing can be said to be passingthrough a closed system. Applying the conservation of mass equa-tion, we can see that the mass flow rate passing through the orificeis equal to the mass flow rate exiting the gap:

min = mout (1)

Apart from the mass conservation equation, the Reynolds equationis also used to determine the pressure profile and the mass flowrate.

∂

∂x

(�h3

12�

∂P

∂x

)+ ∂

∂y

(�h3

12�

∂P

∂y

)= ∂

∂t(�h) (2)

The third dimension, the bearing gap height is relatively small ascompared to the length and width. Therefore, it can be assumedthat there will be no change in pressure across the gap. In order tocalculate the flow rate of air, numerical integration is used to calcu-late the flow rate through the orifice and exiting the gap. The massflow rate per unit length exiting the gap is computed by applyingthe Reynolds equation:

mx = − �h3

12�

∂p

∂x(3)

This is similar for the y-direction and the total sum will form themass flow rate exiting the gap. But in the dynamic (perturbation)case, only the perturbation term of the mass flow rate needs to beused for match the flow from the orifice to the gap. On the otherhand, the mass flow rate through the orifice can be computed byusing the following equation:

m = �d4(PS2 − PD

2)256�RT LC

(4)

By matching Eqs. (3) and (4), the exact pressure profile for bothsteady state and perturbation conditions can be achieved. Fordynamic state performance, perturbation has to be taken intoaccount and the following equations are used. The follow twoequations show the pressure and gap height equations which areaffected by the perturbation forces:

P = p0 + p1eiwt (5)

h = h0 + h1eiwt (6)

The subscripts 0 and 1 represent the steady state and dynamic state,respectively. Substituting (5) and (6) in Eq. (2), we get the followingequations:

Steady State:= ∂2

∂x2(p2

0) + ∂2

∂y2(p2

0) (7)

Perturbation :1

12�RT

[h3

0∂2

∂x2Z1 + h3

0∂2

∂y2Z1

]− iω

Z1

p0h0 = iωp0h1

(8)

where Z1 = P0P1.Eq. (7) is a steady state equation which has the form of Laplacian

equation. It can be also noticed that Eq. (8) is a non-homogenousHelmholtz equation [26]. Below is the development process for set-

ting up a FEM code to solve the Eq. (8). Similar technique can be usedto solve Eq. (7). In order to solve Eq. (8), finite element method isused to approximate the solution and formulate the finite elementequations. Galerkin’s method which introduces shape factor and

402 N. Bhat et al. / Precision Engineering 36 (2012) 399– 407

erim

w(s

∇

∇

w

f

L

c

LRzo∫

tdb

f

g

o

|w

|

|

Fig. 2. Exp

eighting function into the equation is applied [27–29]. First, Eq.8) is split into real and imaginary part. This forms two couplingecond order differential equations.

2f1 = −12�ωh0

h30p0

g1 (9)

2g1 = −12�ωh0

h30p0

f1 + 12�ωh1

h30

p0 (10)

here

1 + ig1 = P0P1 (11)

et

= 12�ωh0

h30p0

and d = 12�ωh1

h30

p0 (12)

et W1(x,y) and W2(x,y) be weighting functions. In Weighted-esidual formulation, the weighted integral of residuals is set to beero. With f1(x, y) and g1(x, y) representing approximated solutionf Eqs. (9) and (10):

A

W1|∇|T |∇ f1| + cg1 + dA +∫

A

W2[|∇|T |∇g1| − cf1 − d]dA = 0 (13)

Eq. (13) is integrated to balance the order of Weighting Func-ions and the approximated solutions. After which the equation isiscretized into elemental form by introducing shape functions aselow:

1 ≈ Nif1i + Njf1j + Nkf1k + Nlf1l + · · · (14)

1 ≈ Nig1i + Njg1j + Nkg1k + Nlg1l + · · · (15)

By manipulation these equations, equation of the form below isbtained:

K ||U| = |F | (16)

here |K| is stiffness matrix, |U| is f1 and g1 terms, |F| is force matrix

K | =∫

(|Be|T |Be| + |Ne|T |C||Ne|)dAe (17)

Ae

F | = −∫

Ae

|Ne|T∣∣∣∣ 0

d(x, y)

∣∣∣∣ dAe (18)

ent setup.

where matrices |Be|, |Ce|, |Ne| and |Ne|, are defined in nomenclaturelist.

The load capacity of a bearing is also an important design param-eter. For both the steady and perturbation stages, the load capacityof the bearing can be obtained by integrating the pressure profileunder the gap with respect to the area of the bearing. Hence it canbe defined by the following equation:

w =∫

P dA (19)

As the supply pressure increases, the load capacity of the bear-ing will increase as well. This is shown in the equation above. Byusing the above equation, the pressure distribution can be attainedfor both steady and perturbation states. This pressure is then inte-grated to get load capacity. Using the following equations, thecoefficient of stiffness and damping can be found:

w1 = −(K + iωC) (20)

A customized code was then built based on the skeleton FEAPcode [30,31]. An input file is required for the code developed. First,ANSYS Multiphysics, commercial simulation software, is used togenerate the geometry and meshing of the model. It is then sup-plied to the FEAP code and Coefficient of Stiffness and Damping aresolved with the code. By manipulating the boundary condition, wecan obtain the characteristic of air bearing performance at differentconditions.

3. Experimental setup

An air bearing test rig as shown in Fig. 2 was set up to studythe steady state performance of air bearings. The base plate (2) ismounted on an external metal frame (1). The assembly consists ofa XYZ motion stage (3) which is mounted on the base plate (2). Theassembly of the reaction plate (5) with the pressure transducer isused to map the pressure profile of the air gap. The reaction plate ismounted on the XYZ stage with the help of a pair of C channels (4). Aflow metre is mounted to measure the flow rate of air entering the

air gap, two load cells (1) measure the load capacity of the air bear-ing and four displacement sensors measure the gap height betweenthe bearing (6) and the reaction plate (5). The displacement sensorsare mounted on the air bearing plate (6).

N. Bhat et al. / Precision Engineering 36 (2012) 399– 407 403

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ep

raph 1. Pressure profile (0.1 mm orifice diameter, gap height 15 �m) load cell.

Air is pressurized by a compressor and is fed through the top ofhe bearing. This causes an air gap to form between the surface ofhe bearing and the reaction plate. The assembly of the hydraulicylinder (8) with the spherical bearing (7) through connecting rod9) keeps the bearing in position when compressed air is suppliedo the bearing. The outputs from the different measuring compo-ents are captured by the DAQ system consisting of a hardwareomponent (analogue to digital converter) and a software compo-ent (Labview). The Labview program is used record the data fromifferent sensors and record it onto a excel sheet. The motion stagehen moves according to a Labview program and this will map oneuarter of the pressure profile of the air gap. The orifice diameter is

n the middle of the bearing and because of symmetry; the pressurerofile will be similar for each quarter of the bearing and hence it

s only necessary to map one quarter of the bearing. The surfacerea which is mapped for pressure has a size of 50 mm ×50 mm,hich corresponds to the surface area of a quarter of the

earing.

.1. Comparison of results

An example of a pressure profile that will be obtained from thexperiment is shown in Graph 1. The theoretical load capacity andressure values obtained are quite similar to the experimental ones.

Graph 2. Results comparison, load capacity.

Graph 3. Results comparison, pressure distribution along the length of the bearing.

Graph 2 shows the load capacity comparison for theoretical andexperimental values for 0.5 mm orifice diameter. As expected theload capacity decreases as the gap height increases since the ori-fice pressure drops off rapidly with the increase in gap height. Thepressure (guage) values are observed to be approaching zero as themeasuring point approaches the end of the bearing since the airescaping the boundary enters the atmosphere. This can be observedin Graphs 3 and 4. The differences observed in the graphs can beattributed to experimental errors such as inconsistent pressuresupply, slight tilting of bearing. Some of the experimental errorsobserved are inconsistent pressure supply, slight tilting of the airbearing, manufacturing errors, etc. Also the apparent difference inthe orifice pressure in Graphs 3 and 4 is due to positioning of thepressure sensor in relation to the orifice. The pressure distributionin the gap is obtained by scanning the area under the bearing start-ing from the centre of the bearing. If the pressure sensor (mountedon the reaction plate) as shown in Fig. 2 is positioned exactly in thecentre of the orifice (as is the case of Graph 4) it will be measur-ing the pressure right below the orifice. However if there is a slighterror in positioning (i.e. the centre of the pressure sensor does notcoincide with the orifice centre) as is the case of Graph 3, the peakpressure (pressure downstream of orifice) observed will be lesseras can be seen by comparing Graphs 3 and 4. Overall the theo-retical results compare well with the experimental results both interms of trend and the actual values thereby validating the FEMcode.

4. Dynamic performance characteristics

The parameters which are varied for each analysis are the bear-ing’s orifice diameter, supply pressure, gap height and the amountof perturbation. An additional parameter which has been studied

Graph 4. Results comparison, pressure distribution along the length of the bearing.

404 N. Bhat et al. / Precision Engineering 36 (2012) 399– 407

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Graph 5. Coefficient of stiffness, varying orifice diameter (15 �m).

s the impact of length/width (L/B) ratio on the dynamic charactersics of the bearing.

.1. Case 1 – varying orifice diameter

The orifice diameter of the bearing and the perturbation fre-uency is varied in this analysis while the rest of the parametersre kept constant. The results from this analysis are shown inraphs 5 and 6. Negative stiffness was observed to be present for.05 mm orifice diameter while negative damping is observed formall orifice diameters (smaller than 0.25 mm). At low perturbationrequencies, the coefficient of stiffness is higher at 0.075 mm and.1 mm as compared to the rest while negative damping is observedor small orifice diameters (less than 0.25 mm). At high perturba-ion frequencies, the coefficient of stiffness increases as the orificeiameter increases while the coefficient of damping converges toero. Negative damping present at low orifice diameters is due tohe presence of pneumatic hammer.

For small orifice diameters, the gap resistance becomes insignif-cant in comparison to the orifice resistance as the gap heightncreases. Thus orifice discharge pressure must drops rapidly withncrease in gap height to satisfy the mass flow balance equationSection 3.1). This translates into the bearing having very less loadapacity and almost negligible stiffness at higher gap heights. Thiseduces the ability of the air bearing to sustain external load orhock. Application of perturbation force leads to a sudden decreasen gap height since the gap pressure needs to balance the pertur-ation force. This compresses the air in the gap which results in

sudden increase in gap pressure. When the perturbation force is

eleased thermodynamic work is done against the bearing plate ashe heated air expands and also due to the energy supplied by thexternal gas source and the gap height increases again. Howeverhe bearing plate is prevented from returning to the equilibrium

Graph 6. Coefficient of damping, varying orifice diameter (15 �m).

Graph 7. Coefficient of stiffness, varying gap height (0.25 mm).

position on account of cyclic nature of the perturbation force. Thisleads to a phase difference between the cyclical displacement ofair inside the gap and the applied perturbation force. Thus thevolume of air inside the gap does sustained work on the system.If the frequency of a resonance extracts enough energy from thepressure-fed gas film to sustain that frequency, the resonance willbe sustained as air hammer instability.

4.2. Case 2 – varying gap height

In this analysis, the parameters that are varied are the gap heightand the perturbation frequency. The results are shown in Graphs7 and 8. There is no negative stiffness observed while negativedamping is observed for large gap heights (larger than 20 �m). Aslow perturbation frequencies, the coefficient of stiffness is at a lowconstant value while the coefficient of damping decreases as gapheight increases. At high perturbation frequencies, the coefficientof stiffness decreases as gap height increases while the coefficientof damping converges to zero.

The reason for having negative damping for large gap heightscan be explained with a similar mechanism. As the gap heightincreases, the volume of air trapped within the gap is larger. Whenit undergoes cyclic expansion and compression due to the pressurebuild up (on account of high gap resistance) and the perturbationforce respectively, higher energy is available to sustain the reso-nance frequency (resulting in hammering effect) due to periodiccompression and expansion of this large trapped volume.

4.3. Case 3 – varying supply pressure

The supply pressure and perturbation frequency are variedwhile the remaining parameters are kept constant. The results areshown in Graphs 9 and 10. Negative damping is only observed at

Graph 8. Coefficient of damping, varying gap height (0.25 mm).

N. Bhat et al. / Precision Engineering 36 (2012) 399– 407 405

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Graph 11. Coefficient of stiffness, varying perturbation amount (orifice diameter0.075 mm, gap height 10 �m).

raph 9. Coefficient of stiffness, varying supply pressure (orifice diameter 0.5 mm,ap height 40 �m).

igh supply pressure of 8 bar. At low frequencies, the coefficient ofamping decreases as the supply pressure is increased while theoefficient of stiffness remains at a constant value. At high per-urbation frequencies, the coefficient of stiffness and coefficientf damping increases as the supply pressure increases. Negativeamping shows that there is pneumatic hammer instability.

High gap resistance due to micro size gap height restricts their flowing out of the bearing gap. As the air being trapped withinhe gap, the pressure within the gap height is built up faster withigher supply pressure. At the same time, when the perturbation

orce tries to compress the gap, it is hardly being compressed byhe force. However, due to high pressure air being trapped withinhe gap, small volume compression can contribute significantly toncrease the pressure within the gap. As the result of the built-upressure, the gap expands and cyclic expansion and compressionegins. Thus the cyclic expansion and compression will becomeut of phase with the change in pressure due to high energy withinhe gap. The high supply pressure becomes the source of energyo sustain the resonance frequency which results in pneumaticammering.

.4. Case 4 – varying perturbation amount

In this analysis, the amount of perturbation and perturbationrequency are varied. The amount of perturbation is calculated as

certain percentage of the gap height at which the air bearing isonsidered to be at during simulation. For example, for a perturba-ion of 0.1% for a gap height of 10 �m, the amount of perturbations considered to be 0.01 �m or 10 nm. Air bearings with large ori-

ce diameter (greater than 0.25 mm) are generally not affected byarying the amount of perturbation. Negative damping is observedo be present. The results are shown in Graphs 11 and 12 for

raph 10. Coefficient of damping, varying supply pressure (orifice diameter 0.5 mm,ap height 40 �m).

Graph 12. Coefficient of damping, varying perturbation amount (orifice diameter0.075 mm, gap height 10 �m).

orifice diameter = 0.075 mm. At low perturbation frequency, thecoefficient of stiffness and damping decreases as the perturbationamount increases. Coefficients of stiffness values are quite similarand coefficient of damping values converges to zero at high per-turbation frequencies. With larger amounts of perturbation, the airbearing is subjected to the pneumatic hammer instability whichresults from the periodic expansion and compression of the airvolume inside the gap. The same trend was observed for orificediameters up to 0.25 mm.

Varying the amount of perturbation at larger orifice diameters(greater than 0.25 mm) has very little impact on the coefficient ofstiffness and damping values. From Graphs 13 and 14, the coef-

ficient of stiffness and damping values are almost similar exceptthe coefficient of damping value at 10% at lower perturbationfrequencies.

Graph 13. Varying perturbation amount (orifice diameter 0.25 mm, gap height15 �m).

406 N. Bhat et al. / Precision Engineering 36 (2012) 399– 407

Fig. 3. Air bearing and hypodermic needle assembly.

G1

4

oaoreasrsd

Gh

assembling hypodermic needle tubing with the air bearing platedescribed in this section can also be used to make commercial off

raph 14. Varying perturbation amount (orifice diameter 0.25 mm, gap height5 �m).

.5. Case 5 – varying L/B ratio

Previously, the analyses were conducted with an L/B ratiof 1. All parameters will be kept constant except the L/B rationd perturbation frequency for this analysis. This analysis isnly conducted for 0.1 mm and 0.5 mm orifice diameters. Theesults are shown in Graphs 15–18. Results from orifice diam-ter 0.5 mm shows a similar trend as seen from Graphs 17nd 18 except that there is no negative damping observed ashown in Graph 18. Negative damping is observed when the L/B

atio is 0.5. At low perturbation frequencies, the coefficient oftiffness shows a low constant value while the coefficient ofamping increases as the L/B ratio increases. At high perturbation

raph 15. Coefficient of stiffness, varying L/B ratio (orifice diameter 0.1 mm, gapeight 10 �m).

Graph 16. Coefficient of damping, varying L/B ratio (orifice diameter 0.1 mm, gapheight 10 �m).

frequencies, the coefficient of stiffness increases as the L/B ratioincreases while the coefficient of damping converges to zero.

5. Manufacturing techniques

Usage of hypodermic needles tubes simplifies the manufactur-ing process as compared to micro-drilling which has an aspectratio. Also the technique of micro drilling can be extremely timeconsuming especially for small orifice diameters. The technique of

the shelf inherently compensated aerostatic bearings. As can be

Graph 17. Coefficient of stiffness, varying L/B ratio (orifice diameter 0.5 mm, gapheight 15 �m).

N. Bhat et al. / Precision Enginee

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raph 18. Coefficient of damping, varying L/B ratio (orifice diameter 0.5 mm, gapeight 15 �m).

een from Fig. 3 the assembly consists of a hypodermic capillaryube which is soldered to a hexagonal cap screw. The cap screw ishen inserted into the bearing through the opening at the top. Afterhe cap screw has been inserted, the opening at the top is sealedith a hexagonal key straight thread O’ ring sealed plug. The bot-

om side of the cap screw is made flush with the bottom side ofhe bearing to ensure that no pocket is present since this can likelyause pneumatic hammer. This design ensures that the capillaryube is easily replaceable if it gets clogged since the hexagonal capcrew can be opened from the top to fit in another hexagonal capcrew hypodermic needle tube assembly. This design also enablesasy assembly and replacement (in case of clogging) of the capillaryube restrictor.

. Conclusion and future work

Pneumatic hammer instability tends to occur at low perturbationfrequencies at small orifice diameters (less than 0.25 mm), largegap heights (greater than 20 �m) and large supply pressures.The amount of perturbation affects only air bearings whose orificediameters are smaller than 0.25 mm.Increasing the L/B ratio of the air bearing will improve its stiffnessand damping characteristics.No negative stiffness is observed for inherently compensated flatpad aerostatic bearings when subjected to dynamic perturbationexcept for the case of very small diameters (0.05 mm or less) athigher gap heights (15 �m or more).Higher frequencies of vibration result in the air trapped in thegap acting like an incompressible fluid rather than a compress-ible fluid, hence no pneumatic hammer is observed for higherfrequencies.The design parameters suggested herein can be used as a guide-line to manufacture inherently compensated air bearings whichare stable under dynamic perturbation forces.So, an optimization method needs to be developed to incorporateall the above factors and performance design criterions.The technique of assembling the hypodermic needle tubing withthe air bearing plate can be exploited to manufacture off the shelfinherently compensated air bearings.

cknowledgments

The authors and the co-authors of this paper would like to thankhe Ministry of Education and National University of Singapore for

[

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ring 36 (2012) 399– 407 407

the FRC funding (WBS No: R265-000-308-112) under which thisproject is being carried out.

References

[1] Khonsari MM, Booser ER. Applied tribology. Wiley-Interscience; 2008. p. 566.[2] New Way Precision. Air bearing application and design guide. New Way Preci-

sion; 2003.[3] Majumdar B, Singh K. Analysis of aerostatic thrust bearings with offset

load. International Journal of Machine Tool Design and Research 1973;13(2):65–76.

[4] Al-Bender F, Brussel H. Symmetric radial laminar channel flow with par-ticular reference to aerostatic bearings. Journal of Tribology 1992;114:630–6.

[5] Khatait J, Lin W, Lin W. Design and development of orifice-type aerostatic thrustbearing. SIMTech Technical Reports 2005;6(1):7–12.

[6] Chen N, Ho K. Performance study of a hydrostatic air thrust bearing. Wear1981;70:207–17.

[7] Al-Bender F, Brussel H. Tilt characteristics of circular centrally fed aerostaticbearings. Tribology International 1992;25(3):189–97.

[8] Fourka M, Tian Y, Bonis M. Prediction of the stability of air thrust bearings bynumerical, analytical and experimental methods. Wear 1996;198:1–6.

[9] Teo C, Spakovszky Z. Modeling and experimental micro hydrostaticgas thrust bearing for micro turbomachine. Journal of Turbomachinery2006;128:597–605.

10] Al-Bender F. On the modelling of the dynamic characteristics of aerostatic bear-ing films: from stability analysis to active compensation. Precision Engineering2009;33:117–26.

11] Kassab S, Noureldeen E, Shawky M. Effects of operating conditions and supplyhole diameter on the performance of a rectangular aerostatic bearing. TribologyInternational 1997;30(7):533–45.

12] Li Y, Ding H. Influences of the geometrical parameters of aerostatic thrustbearing with pocketed orifice-type restrictor on its performance. TribologyInternational 2007;40:1120–6.

13] Mishra A. Analysis if pneumatic instability of an aerostatic rectangular thrustbearing with offset load. Wear 1988;122:1–12.

14] Stout K, Sweeney F. Design of aerostatic flat pad bearings using pocketed orificerestrictors. Tribology International 1984;17(4):191–8.

15] Stout K. Design of aerostatic fat pad bearings using annular orifice restrictors.Tribology International 1985;18(4):200–14.

16] Otsu Y, Miyatake M, Yoshimoto S. Dynamic characteristics of aerostatic porousjournal bearings with a surface restricted layer. Journal of Tribology. Copyright© 2011 by ASME JANUARY 2011, vol. 133/011701-1.

17] Miyatake M, Yoshimoto S. Numericalinvestigationofstaticanddynamiccha-racteristicsofaerostatic thrust bearings with small feed holes. TribologyInternational 2010;43:1353–9.

18] Mukai T. Analysis of dynamic characteristics of air bearing. Nippon Steel Tech-nical Report No. 93; 2006. p. 15–7.

19] Chen YS, Chiu CC, Cheng YD. Influences of operational conditions and geometricparameters on the stiffness of aerostatic journal bearings. Precision Engineering2010;34:722–34.

20] Tala-Ighil N, Fillon M, Maspeyrot P. Effect of textured area on the perfor-mances of a hydrodynamic journal bearing. Tribology International 2011;44:211–9.

21] Li YT, Ding H. Design analysis and experimental study of aerostatic linear guide-ways used in a high acceleration and high precision xy stage. In: Proc. IMechEvol. 221 Part J: Journal of Engineering Tribology; 2007. p. 1–9.

22] Aguirre G, Al-Bender F, Van Brussel H. A multiphysics model for optimiz-ing the design of active aerostatic thrust bearings. Precision Engineering2010;34:507–15.

23] Roa S-K, Kimb S, Kwakb Y, Park CH. A linear air bearing stage with activemagnetic preloads for ultraprecise straight motion. Precision Engineering2010;34:186–94.

24] Bhat N, Barrans SM, Kumar AS. Performance analysis of Pareto optimal bearingssubject to surface error variations. Tribology International 2010;43(November(11)):2240–9.

25] Bhat N, Barrans S, Narasimhan R, Kumar AS, Chuin LT. Static and dynamic anal-ysis of inherently compensated aerostatic flat pad bearings. In: Proceedings ofthe ASME 2010 international mechanical engineering congress and exposition;2010.

26] Hildebrand FB. Advanced calculus for applications. New Jersey: Prentice-Hall;1976. p. 733.

27] Chandrupatla TR, Belegundu AD. Introduction to finite elements in engineering.Prentice Hall PTR; 1997. p. 414.

28] Desai CS, Kundu T. Introductory finite element method. CRC; 2001. p. 496.29] Introduction to finite element method. Pohang University of Science and Tech-

nology; 2005.30] Zienkiewicz OC, Taylor RL, Taylor RL, Zhu JZ, editors. The finite element method.

Butterworth-Heinemann; 2005.31] Zienkiewicz OC, Taylor RL, Taylor RL, editors. The finite element method solid

mechanics. Heinemann: Butterworth; 2000.